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CLEP College Algebra/Sequences and Series

In mathematics, it is important to find patterns. That is what mathematicians do almost everyday of their lives. How they determine patterns is different depending on the type of mathematics they work with. For college algebra, determining patterns is part of the curriculum. The problem below demonstrates one way we determine these patterns. Keep in mind that the problem below is an Exploration and most likely not representative of the types of problems you may see in the CLEP College Algebra exam.

What were we doing in the problem above? Essentially, we were simply trying to find what position some "term" is in. Does it not intrigue you to see math try to find a position of some "term" in a list? We have problems like these as mathematicians because the patterns underlying a "list" of numbers can help us determine new facts of mathematics. After all, what were we doing when using functions? We were trying to find a number using a pattern (the function). Unlike the previous sections, however, we were not given a formula. Luckily enough, it is not difficult to make a formula for a given "list" of numbers. Before we dive into these new problems. It helps to establish definitions.

There is a little disclaimer to get out of the way before we try to solve some problems. First, a sequence can have no pattern. However, for our purposes, we will not count any list of numbers in which no pattern exists. Second, even in simple sequences, numbers of any kind can be named if they follow after a sequence rule. For example, here's a sequence in which the rule is to list prime numbers: $${\left\{ \mathit{2},\mathit{3},\mathit{5},\mathit{7},\mathit{11},\ldots\right\}}$$ If you were like most people, you would probably name the prime numbers in order. However, you could perhaps finish the sequence like this: $${\left\{ \mathit{2},\mathit{3},\mathit{5},\mathit{7},\mathit{11},\mathit{13},\mathit{23},\mathit{2011},\mathit{(2^{82,589,933}-1)}, \ldots\right\}}$$ For our purpose of standardization, we will follow a pattern by stating what number you must find first or identify the pattern.

Let's begin exploring the world that is Sequences and Series.

Sequences
As you already know, a sequence is a list of objects that generally follow a pattern. However, the type of pattern that is described will classify sequences into either arithmetic sequences or geometric sequences. Each will be explored in depth within the next sections.

Arithmetic Sequences
An example could perhaps help you figure with the formal definition above: The sequence $${\left\{ 2, 4, 6, 8, 10, \ldots\right\}}$$ has a one-to-one correspondence with the general sequence $${\left\{ x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, \ldots\right\}}$$ since the first term $$x_{1}=2$$, second term $$x_{2}=4$$, third term $$x_{3}=6$$, and so on. The difference is the amount added to each previous term to get the new term. For $$x_{2}$$, $$x_{1} + d = x_{2}$$ or $$2 + d = 4$$. Solving for $$d$$ is the difference of the two terms. In this example, the difference is $$d=2$$. This is how we define an arithmetic sequence.

Recursive Formula
Often times, we want to generate a sequence using a formula (we are mathematicians, after all, and we like to study sequences to see if there are any general patterns). If we want to find $$x_{i}$$, we may use the following formula:

$$x_{i} = x_{i-1} + d.$$

However, the above formula could describe any sequence that has that general pattern. To fix this, we need to describe the first term as well when using the formula above. There are two ways to describe this formula: x_{1}=a \\ x_{i} = x_{i-1} + d \end{cases}$$
 * 1) We have it horizontally deliniated: $$x_{1}=a; x_{i} = x_{i-1} + d$$.
 * 2) We have it vertically deliniated: $$\begin{cases}

To save space, we will horizontally deliniate formulas for arithmetic sequences in this WikiBook.

Formulas in which the first term is identified along with an equation in which the previous term is added by $$d$$ to get the next term is called a recursive formula.

To determine the next term, we first need to find the difference $$d$$. Note that an arithmetic sequence will have the next term add $$d$$ to the previous term. Since that is how the arithmetic sequence works, $$19+d=18.25$$ is a valid way to find the difference between two terms. Solving for $$d$$, we find the constant difference is $$d=-{3 \over 4}=-0.75$$. Since the difference is the same for each given term in the sequence, we can find the 7th term by adding $$d=-0.75$$ to $$16$$, which gives us $$16-0.75=15.25$$. Finally, add $$d$$ to the next term to get the final answer:

$$15.25-0.75=14.5.$$

There are many reasons why it is more important to have a recursive formula. It is not always slow; it may be easier to understand. The next example shows why this is exactly true

Many of you will perhaps know this famous pattern as the Fibonacci Sequence. For those of you who do not know this sequence, the way we determine the next term is by using the previous terms and adding them together. In our notation, we would say that the term $$a$$ at index $$i$$, $$a_{i}$$ is equivalent to $$a_{i-2} + a_{i-1}$$. Remember, however, we are not done. If a mathematician saw the sequence $${\left\{ 2, 4, 6, 10, 16, 26, 42, \ldots \right\}}$$, he (or she) would determine that $$a_{i}=a_{i-2} + a_{i-1}$$ also describes that sequence. Therefore, we must list the first two terms because listing only the first term would not allow us to get the next term. This means our final answer is

$$a_{1}=1, a_{2} = 1; a_{i}=a_{i-2} + a_{i-1}.$$

Note that Example 1.1.b is not an example of an arithmetic sequence. Your next exploration will be to determine why this is true. Along with that, you will use your critical thinking skills to argue for or against something in the explorations after that one.

Direct Formula
By now, you may be wondering if there is a way to find the term of an arithmetic sequence directly. Well, there is. Before giving you the formula, let us go through the motions for our general recursive arithmetic formula $${x_{1}=a; x_{k}=x_{k-1}+d}$$. Let's chart the recursive arithmetic formula.

If you think about it, the table above is basically a linear function, although starting at $$k=1$$ instead of $$k=0$$. Write out the function as $$x=a+kd$$. We get near our answer. Our independent variable $$k$$ is horizontally translated by $$1$$ to the right, so $$x=a+(k-1)d$$ is our function. In fact, we found our direct relationship. Rewrite it the way we normally write it and we found our direct formula: $$x_{k}=a+(k-1)d$$.

As always, before we can determine the answer, we need to find the "rate of change" of our sequence. Since $$a=22.4$$, we know that $$x_{i}=22.4+(i-1)d$$. The second term is $$29.2$$, so when $$i=2$$, $$29.2=22.4+(2-1)d$$. Of course, now we can solve for $$d$$:
 * $$29.2=22.4+(2-1)d$$;
 * $$29.2=d$$;
 * $$6.8=d$$.

Since we now know the common difference, we can find the smallest $$450$$th term in the sequence. By using our direct formula for an arithmetic sequence, we can find the index in which it is possible. Since $$x_{i}=22.4-6.8(i-1)$$, we find the $$450$$th term by substituting $$i=450$$. Ergo, $$x_{450}=22.4-6.8(450-1)$$. Solve for $$x_{450}$$ to get the final answer.
 * $$x_{450}=22.4-6.8(450-1)$$;
 * $$x_{450}=22.4-6.8(449)$$;
 * $$x_{450}=22.4-3121.2$$;

$$x_{450}=-3030.8.$$

The example above would be a routine, straightforward problem in the CLEP College Algebra exam. However, as practice makes perfect, we will also have non-routine problems that involve thorough understanding of the topic and concepts and skills learned, which will make up 50% of the exam. This is why it is important to do the explorations. While they may not be on a CLEP exam, they are vital in making you think like a mathematician. The next problem will be non-routine problem.

As always, before we can determine the answer, we need to find the "rate of change" of our sequence. Since $$a=22.4$$, we know that $$x_{i}=22.4+(i-1)d$$. We know that the second term is $$21.2$$, so when $$i=2$$, $$21.2=22.4+(2-1)d$$. Of course, now we can solve for $$d$$:
 * $$21.2=22.4+(2-1)d$$;
 * $$21.2=22.4+d$$;
 * $$-1.2=d$$.

Since we now know the common difference, we can find the smallest term needed to reach a negative number in our sequence. By using our direct formula for an arithmetic sequence, we can find the index by solving for $$i$$. Since $$x_{i}=22.4-1.2(i-1)$$, we find the minimal index in which it is possible to have a number less than zero. Ergo, $$0 \geq 22.4-1.2(i-1)$$. Now all we have to do is solve for $$i$$.
 * $$22.4-1.2(i-1) \leq 0$$;
 * $$-1.2(i-1) \leq -22.4$$;
 * $$i-1 \geq {-22.4 \over -1.2}$$;
 * $$i \geq {22.4 \over 1.2} + 1$$;
 * $$i \geq 19.\overline{6}$$.

Since index $$i$$ must be greater than $$19.\overline{6}$$, the minimal index required to find the term that is negative is at

$$i = 20.$$

CLEP Practice Problems: Check Your Understanding
{In the arithmetic series $${\left\{ 2,\,6,\,10,\,14,\ldots,\,398\right\}}$$ in which $$398$$ is the last term, how many terms are in the sequence? { 100 _9 } terms. (Note: typing a comma will give you a decimal, so do not type a comma.)
 * type="{}"}
 * To find the number of terms, first find the constant difference $$d$$ from subtracting the first two values (this is just easier to do). We learn $$d=4$$. Next, since a direct arithmetic formula helps us find any value of index $$i$$, we can find the number of terms by using the direct arithmetic formula: $$x_{i}=a+d(i-1)$$. Since we know that at some value $$i$$, we can find term $$x_{i}=398$$, we simply use those values into our direct arithmetic formula. Therefore, $$398=2+4(i-1)$$ can help us find the number of terms $$i$$, since the last term is $$398$$. With this, after doing all the math, you get $$i=100$$.

{A school district has set-up a system whereby each school week day (days Monday through Friday) is categorized into either an A-day or B-day. Abigail is tired of having her pencils not returned to her. She decided to devise a system in which each student will record their first and last name. The pencils are free for anyone to borrow. However, should a student forget to return a pencil to her before the end of class, a student is charged $$25$$ cents first. After, another $$25$$ cents are charged per cycled block day the pencil is not returned – e.g. if a student accidentally forgot to give back Abigail's pencil, they will be fined the next A-day after the B-day. Joseph had forgotten to return the pencil for $$60$$ school week days. How much does Joseph owe Abigail, according to her system? - $$$15.00$$ - $$$14.75$$ - $$$7.75$$ + $$$7.50$$ - $$$7.25$$
 * type=""}
 * While a student may have used the correct formula, the student did not account for block days. Since there are two (2) block days, once a full cycle passes (A-days and B-days are finished), Joseph is charged $$25$$ cents on top of the original $$25$$ cents. Because the test-taker did not account for this, the answer would have been overestimated.
 * The student did not account for block days. Since there are two block days, once a full cycle passes (A-days and B-days are finished), a student is charged $$25$$ cents on top of the beginning amount of $$25$$ cents. A student may have made another additional mistake. The student did not account for the initial 25 cents. This value is underestimated yet overestimated at the same time.
 * While the student may have correctly divided the number of weekdays in half to get the following answer, he or she may have forgotten to subtract one from her index and multiply it to the difference. Make sure to start with $$$0.25$$ on the sequence instead of $$$0.50$$ cents.
 * The student who answered this correctly deserves a round of applause for paying attention and being careful. Never be too hasty, despite the timer hanging over your head. Always make sure to double check the question and answer before making a hasty jump towards your pseudo-answer. In case the individual guessed, here is an explanation. Since there two different block days (A- and B-block), the number of school days in which the student missed the return of the pencil is greater than the per cycled block day. Therefore, if Joseph borrowed a pencil in B-day but forgot to pay back, he will have to give $$25$$ cents for forgetting to return the pencil and $$25$$ more cents per cycled block days in which Joseph forgot to return the pencil (i.e B-day). As such, an individual will have to pay for the $$30$$ missed days as opposed to the $$60$$ found in the problem. Finally, since an individual must pay $$25$$ cents, the student must be charged first $$25$$ cents, not $$0$$. Along with using the correct formula, the individual can finally finish this problem.
 * Since there two different block days (A- and B-block), the number of school days in which the student missed the return of the pencil is greater than the per cycled block day. If Joseph borrowed a pencil in B-day but forgot to pay back, he will have to give $$25$$ cents and $$25$$ more cents per cycled block days in which Joseph forgot to return the pencil (i.e B-day). As such, an individual starts off with $$$0.25$$ in debt. If the individual did not make this mistake, be sure not to simply multiply $$$0.25$$ by the number of days.

{The following arithmetic sequence is listed: $${\left\{ -{1 \over 3},\,0,\,{1 \over 3},\,{2 \over 3},\,1,\,1{1 \over 3},1{2 \over 3},\,2,\ldots\right\}}$$ Which of the following is the formula that represents the sequence directly using any index $$i$$ to find term $$x_{i}$$? - $$x_{i}={1 \over 3} - {1 \over 3}(i+1)$$ - $$x_{i}={1 \over 3} - {1 \over 3}(i-1)$$ - $$x_{i}=-{1 \over 3} + {1 \over 3}i$$ + $$x_{i}=-{1 \over 3} + {1 \over 3}(i-1)$$ - $$x_{i}=-{1 \over 3} + {1 \over 3}(i+1)$$
 * type=""}
 * The formula used is incorrect for three reasons: [1.] the sequence starts at $$-{1 \over 3}$$, not $${1 \over 3}$$; [2.] the sequence adds $${1 \over 3}$$, not $$-{1 \over 3}$$; finally, [3.] the index term does not start at $$-1$$ (i.e. notice $$i+1$$ instead of $$i-1$$).
 * The formula used is incorrect for two reasons: [1.] the sequence starts at $$-{1 \over 3}$$, not $${1 \over 3}$$; [2.] the sequence adds $${1 \over 3}$$, not $$-{1 \over 3}$$.
 * The formula used is incorrect because the index term does not start at $$0$$ (i.e. notice $$i$$ instead of $$i-1$$).
 * The student is able to successfully make an arithmetic formula when asked.
 * The formula used is incorrect because the index term does not start at $$-1$$ (i.e. notice $$i+1$$ instead of $$i-1$$).

{If the first term is $$x$$ in a sequence, the last $$501$$st term of the sequence is $$9,000$$, and the constant difference is $$20$$, what is the second term of the sequence? - $$-1,000$$ + $$-980$$ - $$-1,020$$ - $$1,000$$ - $$980$$
 * type=""}
 * Be sure not to rush! Notice that the question asked what is the second term, not the first term. Since $$a_{501}=x+20(501-1)=x+10000=9000$$, the first term $$x=-1,000$$. However, to get the second term, one must add 20 to the answer found.
 * Goob job! For any individual that guessed, notice that the question asked what is the second term, not the term before the first one. Since $$a_{501}=x+20(501-1)=x+10000=9000$$, the first term $$x=-1,000$$. However, to get the second term, one must add 20 to the answer found.
 * Be sure not to subtract your answer! Notice that the question asked what is the second term, not the term before the first one. Since $$a_{501}=x+20(501-1)=x+10000=9000$$, the first term $$x=-1,000$$. However, to get the second term, one must add 20 to the answer found.
 * Take a look again at the Algebra! Since $$a_{501}=x+20(501-1)=x+10000=9000$$, the first term $$x=-1,000$$. However, to get the second term, one must add 20 to the answer found.
 * Take a look again at the Algebra! Since $$a_{501}=x+20(501-1)=x+10000=9000$$, the first term $$x=-1,000$$. However, to get the second term, one must add 20 to the answer found.

{Given $$a_{i}$$ is the term of a sequence at index $$i$$, the first term of a sequence is $$x$$, and the constant difference is $$d$$, which of the following must be TRUE? Indicate all such statements. + $$a_{1}=x; a_{i}=x+d(i-2)+d$$ - $${x-a_{i} \over d}-1=i$$ + $$a_{i}-x+d=di$$
 * type="[]"}
 * Note that a sequence can still follow using a recursive formula. As such, knowing that $$a_{i}=x+d(i-1)$$, $$a_{i-1}=x+d([i-1]-1)=x+d(i-2)$$. In which case, $$a_{1}=x; a_{i}=a_{i-1}+d$$ which is equivalent to $$a_{1}=x; a_{i}=x+d(i-2)+d$$.
 * Note that the normal arithmetic sequence is $$a_{i}=x+20(i-1)$$. If one wants to find index $$i$$, then subtract $$x$$, divide by $$20$$, and $$1$$ to yield $${a_{i}-x \over 20}+1=i$$.
 * Note that the normal arithmetic sequence is $$a_{i}=x+20(i-1)$$. Since you multiply the constant difference $$20$$ to $$i-1$$, distribute it to each term in the parenthesis. Given $$a_{i}=x+20(i-1)=x+20i-20$$, try to isolate $$20i$$ on the right side by subtracting $$x$$ and adding $$20$$: $$a_{i}-x+20=20i$$.

Geometric Sequence
As always, if you are unable to understand, try a few examples of numbers to think of in your head. Let the common ratio $$r=2$$ and let $$x_{1}=1.5$$. The next term $${x_{2}=x_{1}r}$$, so $$x_{2}=1.5\cdot2=3$$. If you keep the pattern going for each term of the sequence, you would get the following: $${\left\{ 1.5, 3, 6, 12, 24, 48,\ldots \right\}}$$

Recursive Formula
As with the arithmetic formula, you can find the recursive formula and the direct formula for a geometric sequence. Since every term is multiplied by common ratio $$r$$, let any term of index $$i$$ be represented by $$x_{i}$$. To find the next term requires knowing the previous term. Ergo, $$x_{i}=rx_{i-1}$$

Direct Formula
As with the arithmetic sequence, Let us chart the recursive geometric formula. $${x_{1}=a; x_{k}=rx_{k-1}}$$.

If you think about it, the table above is basically an exponential function, $$f(k)=ab^{k}$$, starting at $$k=1$$. Write out the function as $$x=ar^k$$. We get near our answer. Our independent variable $$k$$ is translated to the right $$1$$ unit, so $$x=ar^{k-1}$$ is our function. In fact, we found our direct relationship. Rewrite it the way we normally write it and we found our direct formula: $$x_{k}=ar^{k-1}$$.

To find the answer, we need to know the common ratio $$r$$ of the sequence above. Pick any arbitrary term in the sequence and apply it to the direct geometric formula: $$x_{2}=ar^{2-1}$$. Find $$r$$:
 * $$11.2=22.4r.$$
 * $$r=0.5={1 \over 2}.$$

Knowing the value of $$r$$, you can find the $$18$$th term in the sequence by using the direct formula:
 * $$x_{18}=22.4 \left ( \frac{1}{2} \right ) ^ {18 - 1}$$
 * $$x_{18}=22.4 \left ( 0.00000762939 \right )$$

$$x_{18}=0.00017089843$$

As always, these examples are things you can work through yourself or follow along so that you can see how to do a problem.

Series
It is great to find these patterns to these sequences, but is that the only use we have for these sequences? As always, the answer in math is never no in regards to a low amount of utility (i.e. usefulness). A function does not only describe the pattern associated with numbers but also predicts the graph created when plotted using $$x$$ for its inputs and $$y$$ for its outputs. Using the terms of a sequence, can we determine a sum? We can, and it is called a series.

There is always a way to find the sum of a sequence by force: add up the terms one by one and get the answer. Mathematicians are lazy people and don't want to do more work than necessary. That is to say, mathematicians work smart, not hard.

Arithmetic Series
This is a classic example among many mathematicians. You may be reaching for your calculator, but that will definitely take a long time to punch in all those numbers, and you may accidently make mistakes while writing out the expression. The question is, how do we solve this in the short amount of time we are given on the CLEP exam? Notice that there is a pattern at play: the first term of the series, $$1$$, plus the last term of the series, $$100$$, will give us $$101$$. The second term of the series, $$2$$, plus the second-to-last term of the series, $$99$$, will give us $$101$$. In fact, for all $$100$$ terms in the series, each pair of numbers, according to their "placement," adds to $$101$$. Since there are $$50$$ pairs in this series, and the sum of each pair of numbers always yields $$101$$,

$$101\cdot50=5050 \text{ is the sum of the expression above.}$$

Let us make a conjecture for the above statement. After all, we are not robots that are simply fed a method and do something the same way. We will contract parts of the above expression in the question above to see if our method will be useful for any number of terms. Let $$n$$ be the number of terms in the arithmetic series $$A_{E}$$, let $$S$$ be the sum of $$A_{E}$$, and let $$S_{m}$$ be the column in which the same method we used in Example 2.1.1.a will, hopefully, get us the sum $$S$$.

Notice how the odd terms have this weird (or perhaps odd) behavior where not all pairs of terms have the same sum (because there is not an even amount of pairs). While it may seem that using the method in Example 2.1.1.a will work for all odd number of terms based on the table above, maybe it is not true for some really high odd term. Therefore, it is important to prove this is true. For now, let us simply define the formula to be true for both even and odd number of terms.

A finite set of things is something that has a determined number of anythng within that set. An infinite set is a set not described by a determined number of the amount within that set.

Note: the above formula is not known as the "Gaussian method" among mathematicians. This wikibooks will simply refer to this method as the "Gaussian method."[see footnote 1.]

This seems like an impossible problem to someone who does not know about the Gaussian method or the properties of an arithmetic sequence. However, since you have paid attention, you can figure out for yourself. For the purposes of giving an example, this Wikibooks will explain. We do not know how many terms there are in the sequence, so let's find out how many terms there are. Since $$S_{A}={a+x_{k} \over 2}\cdot k,$$

$$600={50+250 \over 2}\cdot k.$$

This subject is not called College Algebra for nothing, so do some algebra. $$600={300 \over 2}\cdot k$$

$$600=150k$$

$$40=k$$

Note, however, that we are not looking for the number of terms in the series. We want to know the constant difference of the arithmetic series. Therefore, use the arithmetic sequence direct formula: $$x_{k}=a+d(k-1)$$! $$x_{40}=250$$

$$250=50+d(40-1)$$

$$200=39d$$

$${200 \over 39}=d$$

$$5{5 \over 39}=d$$

Example 2.2.1.b helps us know how to find the number of terms and the constant difference. Sometimes, you may simply not know one or both of the information you needed to find in Example 2.2.1.b., more often the number of terms. The next example helps illustrate the usefulness of knowing about the properties of arithmetic series and arithmetic sequences.

To find the sum, we need to know how many terms there are; otherwise we cannot use the Gaussian method. Because we do not know the sum of the series, let's use the direct arithmetic sequence formula (since we know which terms correspond to place). We want to find the number of terms, $$k$$, so use the last term ($$x_{k}=2,504$$).

First, find the constant difference. To get from $$4$$ to $$9$$, we need to add $$5$$ to $$4$$. Therefore, the constant difference is $$d=5$$. To put it more formally: if recursive formula $$x_{1}=4; x_{2}=4+d$$ and $$x_{2}=9$$, then $$x_{1}=4; 9=4+d$$ means $$d=5$$.

Second, find the number of terms. If $$x_{k}=2,504$$, $$d=5$$, and $$x_{1}=a=4$$, then $$2,504=4+5(k-1)$$. Solve for $$k$$:

$$2,500=5(k-1)$$

$$500=k-1$$

$$501=k.$$

Finally, use the Gaussian Method:

$$S_{A}={4+2,504 \over 2}\cdot 501$$

$$S_{A}={2,508 \over 2}\cdot 501$$

$$S_{A}=1,254\cdot 501$$

$$S_{A}=628,254.$$

As you can see, an arithmetic series can be useful in describing any type of sequence. Now that we have sufficiently explored arithmetic series, let's prove that the Gaussian method works for odd number of terms.

''Note: what you are about to learn is NOT required for the curriculum. If you do not understand this proof, do not worry, for it does not matter for the CLEP exam. These proofs are only to build a mathematical understanding of the concepts. As such, you may skip these if you want.''

CLEP Practice Problems: Check Your Understanding
{Given the arithmetic series $$50+45+30+\ldots-700$$, determine the number of terms in the arithmetic series. { 151 _5 }. (Note: typing a comma will give you a decimal, so do not type a comma.)
 * type="{}"}
 * To find the number of terms in the arithmetic series,

{Given the arithmetic series $$50+45+30+\ldots-700$$, determine the sum of the arithmetic series. { -49075|-49075 _8}. (Note: typing a comma will give you a decimal, so do not type a comma.)
 * type="{}"}

{Function $$S$$ has the following properties: $$S(x)=x\text{ if }-20\le x\le50$$ and $$S(x)=3x+1\text{ if }50<x\le200$$. A student decided to add all terms for all $$x\in\mathbb{Z}$$ or for all x that belong to the set of integers. What is the sum of the terms of function $$S$$ for all $$x\in\mathbb{Z}$$? - $$56,625$$ - $$57,641$$ + $$57,690$$ - $$57,450$$ - $$56,776$$
 * type=""}
 * You were so close! You forgot to add $$1,065$$ to the sum. For the individual who guessed, know that this question was difficult. First, given that $$S(x)=x\text{ if }-20\le x\le50$$, since adding all $$x$$ for this piece of the function is equivalent to $$-20-19-18-\ldots+48+49+50$$, $${-20+50 \over 2}k$$ is the sum of that piece. To find the number of terms in that sequence, use the direct arithmetic formula: knowing $$x_{k}=50$$, $$50=-20+(k-1)$$. Solve for $$k$$ to get $$71=k$$. Now find the sum: $${-20+50 \over 2}\cdot(71)=1,065$$. Do the same process for the other piece. Simply substitute those values of $$x$$ to that piece: $$3(51)+1=154$$ and $$3(200)+1=601$$. Notice that the constant difference is $$3$$ for every term, because $$3(52)+1=157$$, and $$157-154=3$$. Since we do not know how many terms there are, solve for $$k$$: $$601=154+3(k-1)\implies 447=3(k-1)\implies 149=k-1\implies 150=k$$. Find the sum: $${154+601 \over 2}\cdot(150)=56,625$$. Finally, add the two sums of each piece: $$56,625+1,065=57,690$$.
 * You got some part of it right, but some other part of it wrong. While you were able to determine the sum for all $$-20\le x\le50$$, you mistakenly used $$50\le x\le200$$ instead of $$50<x\le200$$. For the individual who guessed, know that this question was difficult. First, given that $$S(x)=x\text{ if }-20\le x\le50$$, since adding all $$x$$ for this piece of the function is equivalent to $$-20-19-18-\ldots+48+49+50$$, $${-20+50 \over 2}k$$ is the sum of that piece. To find the number of terms in that sequence, use the direct arithmetic formula: knowing $$x_{k}=50$$, $$50=-20+(k-1)$$. Solve for $$k$$ to get $$71=k$$. Now find the sum: $${-20+50 \over 2}\cdot(71)=1,065$$. Do the same process for the other piece. Simply substitute those values of $$x$$ to that piece: $$3(51)+1=154$$ and $$3(200)+1=601$$. Notice that the constant difference is $$3$$ for every term, because $$3(52)+1=157$$, and $$157-154=3$$. Since we do not know how many terms there are, solve for $$k$$: $$601=154+3(k-1)\implies 447=3(k-1)\implies 149=k-1\implies 150=k$$. Find the sum: $${154+601 \over 2}\cdot(150)=56,625$$. Finally, add the two sums of each piece: $$56,625+1,065=57,690$$.
 * The person who got this right deserves a round of applause because this question is trickier than you thought at first glance. For the individual that guessed, know that this question is a little more difficult than you most likely expected. First, given that $$S(x)=x\text{ if }-20\le x\le50$$, since adding all $$x$$ for this piece of the function is equivalent to $$-20-19-18-\ldots+48+49+50$$, $${-20+50 \over 2}k$$ is the sum of that piece. To find the number of terms in that sequence, use the direct arithmetic formula: knowing $$x_{k}=50$$, $$50=-20+(k-1)$$. Solve for $$k$$ to get $$71=k$$. Now find the sum: $${-20+50 \over 2}\cdot(71)=1,065$$. Do the same process for the other piece. Simply substitute those values of $$x$$ to that piece: $$3(51)+1=154$$ and $$3(200)+1=601$$. Notice that the constant difference is $$3$$ for every term, because $$3(52)+1=157$$, and $$157-154=3$$. Since we do not know how many terms there are, solve for $$k$$: $$601=154+3(k-1)\implies 447=3(k-1)\implies 149=k-1\implies 150=k$$. Find the sum: $${154+601 \over 2}\cdot(150)=56,625$$. Finally, add the two sums of each piece: $$56,625+1,065=57,690$$.
 * Note that your answer is very wrong. You may have used the correct numbers for $$S(x)=x\text{ if }-20\le x\le50$$, but the individual determined that there were 70 terms instead of 71 (make sure to include the first term). Second, make sure to use the correct numbers for $$S(x)=3x+1\text{ if }50<x\le200$$. Use all numbers above 50. For the individual that guessed, know that this question is a little more difficult than you most likely expected. First, given that $$S(x)=x\text{ if }-20\le x\le50$$, since adding all $$x$$ for this piece of the function is equivalent to $$-20-19-18-\ldots+48+49+50$$, $${-20+50 \over 2}k$$ is the sum of that piece. To find the number of terms in that sequence, use the direct arithmetic formula: knowing $$x_{k}=50$$, $$50=-20+(k-1)$$. Solve for $$k$$ to get $$71=k$$. Now find the sum: $${-20+50 \over 2}\cdot(71)=1,065$$. Do the same process for the other piece. Simply substitute those values of $$x$$ to that piece: $$3(51)+1=154$$ and $$3(200)+1=601$$. Notice that the constant difference is $$3$$ for every term, because $$3(52)+1=157$$, and $$157-154=3$$. Since we do not know how many terms there are, solve for $$k$$: $$601=154+3(k-1)\implies 447=3(k-1)\implies 149=k-1\implies 150=k$$. Find the sum: $${154+601 \over 2}\cdot(150)=56,625$$. Finally, add the two sums of each piece: $$56,625+1,065=57,690$$.
 * Note that your answer is very wrong. You may have used the correct numbers for $$S(x)=x\text{ if }-20\le x\le50$$, but the individual determined that there were 70 terms instead of 71 (make sure to include the first term). Second, make sure to use the correct numbers for $$S(x)=3x+1\text{ if }50<x\le200$$. Use all numbers above 50. Finally, if you wanted to find the sum of the two pieces, then you had to add. For the individual that guessed, know that this question is a little more difficult than you most likely expected. First, given that $$S(x)=x\text{ if }-20\le x\le50$$, since adding all $$x$$ for this piece of the function is equivalent to $$-20-19-18-\ldots+48+49+50$$, $${-20+50 \over 2}k$$ is the sum of that piece. To find the number of terms in that sequence, use the direct arithmetic formula: knowing $$x_{k}=50$$, $$50=-20+(k-1)$$. Solve for $$k$$ to get $$71=k$$. Now find the sum: $${-20+50 \over 2}\cdot(71)=1,065$$. Do the same process for the other piece. Simply substitute those values of $$x$$ to that piece: $$3(51)+1=154$$ and $$3(200)+1=601$$. Notice that the constant difference is $$3$$ for every term, because $$3(52)+1=157$$, and $$157-154=3$$. Since we do not know how many terms there are, solve for $$k$$: $$601=154+3(k-1)\implies 447=3(k-1)\implies 149=k-1\implies 150=k$$. Find the sum: $${154+601 \over 2}\cdot(150)=56,625$$. Finally, add the two sums of each piece: $$56,625+1,065=57,690$$.

{The first term of an arithmetic sequence is $$a<0$$. Let there be two separate sequences, $$\left \{ x_{i} \right\}=a+d(i-1)$$ and $$\left \{ a_{k} \right\}=a+d(k-1)$$, with $$500$$ terms in each sequence. If the constant difference is $$d<0$$ for $$\left \{ x_{i} \right\}$$, but the constant difference is $$d>0$$ for $$\left \{ a_{k} \right\}$$, which of the following must be true? Indicate all such statements + $$x_{1}-x_{2}-\ldots-x_{499}-x_{500}-a_{1}-\ldots-a_{499}-a_{500}=998a$$ + $$x_{1}+x_{2}+\ldots+x_{499}+x_{500}+a_{1}+\ldots+a_{499}+a_{500}=-1,000a$$ + $$x_{500}<0$$
 * type="[]"}

Note, this practice quiz is unfinished.

Proof of Arithmetic Series
''Note: what you are about to learn is NOT required for the curriculum. If you do not understand this proof, do not worry, for it does not matter for the CLEP exam. These proofs are only to build a mathematical understanding of the concepts. As such, you may skip these if you want.''

Geometric Series
While it is amazing one can find the sum of an arithmetic sequence, what about the sum of a geometric sequence? Thankfully, mathematicians have found a way to calculate this idea. This theorem (not conjecture, as you will hopefully see with the geometric series proof) is one of many ways mathematicians have found solutions to problems coming from the abstract. As always, it is best to understand the way you can use a formula first before getting the formula introduced. To invoke deeper learning, we want to understand the concept, not reciprocate the formula outloud on a test and say, "I know the answer."

Let's define a sum $$S$$ for the geometric sequence: $${1+2+4+8+\ldots+512=S}$$

Notice that all the terms except $$x_{1}=1$$ has a factor $$2$$. Therefore, move the first term to the other side by subtracting $$1$$: $${2+4+\ldots+512=S-1}$$. Then, factor the two on the left side: $${2(1+2+4+\ldots+256)=S-1}$$

The key discovery of this exercise is noticing that $${1+2+4+\ldots+256}$$ is $$S$$ but without $$512$$, so given that $${S=1+2+4+8+\ldots+512}$$,

$${1+2+4+\ldots+256}=S-512$$

Solve using elementary algebra:

$${2S-1024=S-1}$$

$${2S-S=-1+1024}$$

$${S=1023}$$

Notice how we solved the problem. We said that the sum must equal something positive, so we determined that if it does equal something, we could figure it out in some way without having to do any long calculations. Can we use this method for some general geometric sequence? This how we determine a formula. In your Exploration, you will be tasked with presenting a proof for the formula of the geometric series. We will still give you the formula, but you must prove it yourself in the next exploration. Of course, we will also present another way to prove the geometric series formula in the next section.

Proof of Geometric Series
''Note: what you are about to learn is NOT required for the curriculum. If you do not understand these concepts, do not worry, for it does not matter for the CLEP exam. These proofs are only to build a mathematical understanding of these concepts. As such, you may skip these if you want.''

Sigma Notation
How many of you are tired of writing out the expressions? How many of you are tired of figuring out how many expressions there are in a series? Where is this short-hand that we conveniently have for an arithmetic or geometric sequence? Thankfully, all of your answers are coming soon.

Say you want to write out the expression $$\underbrace{1+1+1+1+\ldots+1+1}_{500}$$, in which there are $$500$$ terms in the series, each of which are the number $$1$$. We hopefully can evaluate the series without having to think too much about it, or even think about a short-hand about this expression immediately without too much thought. However, the purpose of this easy exercise is to introduce a new notation.

Let $$\sum$$ denote the sum of a series. Located at the bottom of the symbol is the start of the series at index $$i=a$$ (blue) and the top of the symbol is the last index of the series, $$k$$ (red). The term inside the parentheses represents the sequence that the series follows, $$f(i)$$ (orange). The use of the notation of is shown below $$\sum_{ {\color{Blue}i=a} }^{ {\color{Red} k} } \left( {\color{Orange} f \left( i \right) } \right) .$$Here, we know that $$\left\{ 1, 1, 1, 1, \ldots 1, 1 \right\}$$ must have the sequence $$x_{i}=1 + 0(i - 1) = 1=f\left( i \right)$$.

Interpreting the meaning of $$f(i)$$ tells us that for any number in the sequence, the $$i$$th term of the expression is $$1$$. Knowing that the sequence starts at index $$1$$, meaning $$i=1$$, and there are $$500$$ terms, meaning $$k=500$$, we may write the sigma notation as below: $$\sum_{ {\color{Blue}i=1} }^{ {\color{Red} 500} } \left( {\color{Orange} 1 } \right) .$$

It is important to write the sigma representation having $$f(i)$$ in parenthesis. Often, if you don't have it, you can confuse other terms by accident. It is for this reason that this Wikibooks recommends writing the $$f(i)$$ term in parentheses.

Some more ways in which sigma notation is written is often a short-hand, especially when handwriting. This should be used if you want to save time.
 * $$\sum_{i=a}^{k}\left(f(i)\right) = \sum_{a\le i\le k} \left(f(i)\right)$$
 * $$\sum_{i=a}^{\infty}\left(f(i)\right) = \sum_{i\ge a} \left(f(i)\right)$$

Before we jump into our exploration, it may be a good idea to introduce some more expressions that can be rewritten into its sigma form equivalents. Some of the expressions may not even follow from a formula we have seen so far. Nevertheless, let us continue with this concept.

A good idea before getting started is to look for any possible patterns. Because you are working with sigma notation, check whether there has to be a pattern involving addition.

Look at the numerator of the first two terms. Let us assume we are starting at index $$i=1$$. To get from the first term in the numerator to the second term in the numerator, one simply has to add 1 to the first term. Similarly, to get from the first term in the denominator to the second term in the denominator, one simply has to add 2 to the first term. Both of these are true on a term-by-term basis.
 * $$f(i)=\frac{i+1}{i+2}$$. This is easy to verify for every term in its respective index.

Because this function is true, let us use this to find the number of terms in the expression (i.e., look at the final term in the expression).
 * Let $$f(k)=\frac{71}{72}=\frac{k+1}{k+2}$$. There are many ways to solve for $$k$$, the last term of the expression, in this equation. We will show the standard way.
 * $$\frac{k+1}{k+2}=\frac{71}{72}$$
 * $$\Leftrightarrow72\cdot(k+1)=71\cdot(k+2)$$
 * $$\Leftrightarrow72k+72=71k+142$$
 * $$\Leftrightarrow k=70$$

We have demonstrated this expression involves 70 terms. With the index, number of terms, and function ready, we may now write the sigma equivalent: $$\sum_{i=1}^{70}\left(\frac{i+1}{i+2}\right)\blacksquare$$ Alternatively, one can change the index to get a completely different sigma notation. Notice how when the index starts at an $$i>1$$, the formula and the final term's index also changes. Here, we will look specifically at $$i=2$$. Keeping in mind the index represents the placement along the series, going from 2 to 3 is simply a matter of changing along with the numerator. Because the denominator is simply the same idea except displaced by 1 more than the index, the denominator follows a pattern of $$i+1$$. Therefore, the function of the sigma notation is
 * $$g(i)=\frac{i}{i+1}$$. This is easy to verify for every term in its respective index.

Use this function to find the the final term's index.
 * Let $$f(k)=\frac{71}{72}=\frac{k}{k+1}$$. We will solve this one the simplest way. Because both the numerator and the denominator are constant, non-zero terms, and $$k$$ is simply an expression of the index for its final term, one can simply set the numerator equal to the other numerator, and the same is true for the denominator. That is,
 * $$\begin{cases}

k=71 \\ k+1=72 \end{cases}$$
 * This is perhaps the easiest systems of equations one had to deal with this year learning college algebra. From this, it is obvious $$k=71$$.

Knowing the index starting point, function, and final index, the sigma form of this situation is $$\sum_{i=2}^{71}\left(\frac{i}{i+1}\right)\blacksquare$$ Notice how the pattern changed. Quite intriguing, would you not say?

Some expressions are too hard to determine simply from looking. These are likely going to involve factors multiplied to another. THe best way to determine such a pattern would be to divide terms to see what the factors are.

There seems to be no pattern here. The best thing to do is to determine any similarities between terms and see if there seems to be a pattern from there.

Notice how the first and fourth term are zero. This means that there is a term that allows us to obtain a value of zero. Let us assume they are linear functions. If $$k=0$$ is the index of the first term and $$k=3$$ is the third term, then a possible summation function is $$\sum_{k\ge0} \left(k(k-3)\right)$$. Let us apply this and see if it works. It seems this function works for each term. Therefore, this works.
 * $$k=0\Rightarrow 0$$
 * $$k=1\Rightarrow -2$$
 * $$k=2\Rightarrow -2$$
 * $$k=3\Rightarrow 0$$
 * $$k=4\Rightarrow 4$$
 * $$k=5\Rightarrow 10$$
 * $$\sum_{k=0}^{\infty} \left(k(k-3)\right)=\sum_{k=0}^{\infty} \left(k^{2}-3k\right)\blacksquare$$

More examples will be added later. The next exploration will ask you write out a variety of expressions using sigma form:

Sigma Simplification Techniques
In some textbooks, this section would be called rules. We call them what they really are: techniques to simplify common sigma representations.

As always, if you are confused, write out some terms. The above sigma representation states that for any constant term $$c$$ added $$k$$ times, the resulting sum is equivalent to $$\underbrace{c+c+c+\ldots+c+c+c}_{k}=ck$$. Never forget that multiplication is repeated addition. This axiom you heard in elementary school is still important to this day.

This second "rule" can be simplified as the following. Since $$\sum_{i=1}^{k} \left( ci \right)$$, we can conclude that $$\sum_{i=1}^{k} \left( ci \right)=1c+2c+3c+\ldots+c(k-2)+c(k-1)+ck$$. Each term in the expression $$1c+2c+3c+\ldots+c(k-2)+c(k-1)+ck$$ has common factor $$c$$, so $$c(1 + 2 + 3 +\ldots+(k-2)+(k-1)+k)$$. Notice that each term on the inside of the parenthesis is a basic series where $$i$$ starting at $$1$$ goes to final term $$k$$ as a sum, so

$$\sum_{i=1}^{k} \left( ci \right)=c\sum_{i=1}^{k} \left( i \right).$$

This one is harder to see why it is true. As always, however, try it out by hand. Remember that sigma notations are short hands of sums that have some sort of formula, so write out a term in case you are ever confused. Notice that $$\sum_{i=1}^{k} \left( c+i \right)=(c+1)+(c+2)+(c+3)+(c+4)+\ldots+(c+k-1)+(c+k)$$. Because adding is commutative, and because the parentheses do not change the sum of the series, you may group terms such that $$(c+1)+(c+2)+(c+3)+(c+4)+\ldots+(c+k-1)+(c+k)=\underbrace{c+c+c+c+\ldots+c+c}_{k} + (1+2+3+\ldots+k-1+k)$$ We have already determined the sigma notation for each grouping, so we may put it together to say the following is true:

$$\sum_{i=1}^{k} \left( c+i \right)=kc+\sum_{i=1}^{k} \left( i \right).$$

Realize that the above "rule" is simply an extension of the previous one. If $$f(i)=i$$, then $$\sum_{i=1}^{k} \left( c+f\left( i \right) \right)=kc+\sum_{i=1}^{k} \left( f\left( i \right) \right)$$. Realize that any function can comply with this rule. By extension:

$$\sum_{i=1}^{k} \left( cf\left( i \right) \right)=c\sum_{i=1}^{k} \left( f\left( i \right) \right)$$

Using this general knowledge, one can also argue the next following "rule":

Given the way we found the same identities, you may do the proof for this one by yourself as an exercise in sigma notation. The best way to understand this new notation is to practice it. (Of course, this Wikibooks will provide plenty of practice.)

Along with that practice, you may also try to show that the identity for the next "rule" is also true:

With this, you now have a sufficient foundation for the necessary tools needed to prove and disprove statements as well as create your own identity. Before we move on the next section, we must mention the sigma identity for each of the series type we learned in this wikibooks.

Sigma Identities for Series Types
What is the sigma identity of an arithmetic series? Before we give you the identity, it is important to understand arithmetic series. If the person reading this wikibooks has jumped from one section to the other, we may recommend you read anything from the previous if the current reasoning seems to not make sense. First, how do we write an arithmetic sequence using a formula? Like this: $$x_{k} = a + (k-1)d.$$ The formula for how to find an arithmetic series is this: $$S_{a}={a + x_{k} \over 2} k.$$ Notice how $$x_{k}$$ is in both formulas, so substitute $$x_{k}$$ for $$a+(k-1)d$$ and you get: $$S_{a}={2a + (k-1)d \over 2} k = ak + {(k-1)d \over 2} k = \left[ 2a + (k-1)d \right] {k \over 2}.$$ Because $$x_{k} = a + (k-1)d.$$ is also a formula that helps us find each term in the series, we have finally created our sigma identity:

We can use the same process to find the sum of a geometric series. You may consider this your next exercise proving that the sigma form of the geometric series is the same one shown below.

Check Your Understanding
The problems below are basically comprehension questions. If you can do all the problems below, you understand all you need to know about sigma notations. Keep in mind, these problems are much harder than usual CLEP problems; however, being able to do these problems with proficiency proves understanding of the material at a deeper level, which means you are better prepared for easier problems and harder problems down the line.

Infinite Geometric Series
There is this old and famous paradox that perplexed many who listened to it. The problem stems from the Greek thinker Zeno of Elea. This old adage was the first example of reductio ad absurdum (disproving a statement by showing that the application of the statement will lead to a contradiction, and so the original statement cannot be true). However, today, a version is no longer a paradox, and it took the invention of calculus to prove this enigma is not an enigma[see footnote 3].

The version of the problem that demonstrates the enigma is proven and shown below.

Note: the paradox will be shown soon.

The age-old questions of most students in mathematics arise: "When are we going to need this?" The next exploration gives one problem in which everything you have learned will be tested. This is definitely harder than many CLEP exam questions; however, problems are what make you better at math. You learn as you do, and so you must.

Proof of Infinite Geometric Series
$$\sum_{i=1}^{\infty} \left( ar^{i-1} \right)={a \over 1 - r}\text{ if }0< \left| r \right \vert < 1$$. How do we prove this is true? Again, this is way above most curriculum required textbooks. However, the goal of many mathematics textbooks is to get the user inspired in math and to use those concepts into other fields of interest. No matter what given field a person is going into, they are going to need to solve problems: a history major needs to understand what a person meant; an English major needs to find the right words that help them convey their ideas; a Science major will have to apply mathematical concepts to communicate ideas about science. As such, learn to love problems, for conquering a problem will make you stronger.

''Note: what you are about to learn is NOT required for the curriculum. If you do not understand these concepts, do not worry, for it does not matter for the CLEP exam. These proofs are only to build a mathematical understanding of these concepts. As such, you may skip these if you want.''

Prerequisite for the proof: Imagine we want to find values of some function $$f(x)$$. Define this function as $$f(x)={x^2 - 4 \over x - 2}$$. If $$x=2$$, then $$f(2)={(2)^{2} - 4 \over (2) - 2}={0 \over 0}$$. Well, it seems this value is undefined when $$x=2$$, so it seems it does not equal anything. However, let's try graphing this rational function.

First, let us divide: $${x^2 - 4 \over x-2} = {(x-2)(x+2) \over x-2} =x+2$$. It seems once we simplify the rational expression, we are left with $$x+2$$. However, This is not exactly true because our function is still undefined at $$x=2$$, so we must add a constraint: $$f(x)=x+2\text{ if }x\ne2$$. To put it more formally:

$$f(x) = \begin{cases} x+2, & \text{if} & -\infty<x<2 \\ {x^2 - 4 \over x-2}, & \text{if} & x=2\\ x+2, & \text{if} & 2<x<\infty \end{cases}$$

If we graph the relation, we find that there is a "hole" at that point in the function. Let's use values close to 2 so that we abide by those rules.

It becomes clear that as values get "closer and closer" to $$2$$, $$f(2)$$ "approaches" $$4$$. What is "close"? How close are your eyes to the screen? We would most likely say, "it is 45 cm. away from the screen." In which case, let's use the distance (i.e. absolute value) to define the word "close": $$|2.001 - 2|=0.001$$ and $$|1.999-2|=0.001$$. (Answer this question: why did we use absolute values?)

Let's define this process using two variables $$x_1$$ and $$x$$: If $$0< \left| x_{1} - x \right\vert <n$$, then those two values are "close" or "approaches $$L$$ of function $$f(x)$$," where $$n$$ can be any value you want it to be, as long as it tolerates our definition. We can conclude the following: $$\left | f(x) - L \right \vert$$ is "close" when $$\left | x_{1} - x \right \vert$$ is close, given that $$0< \left | f(x) - L \right \vert <n_{f}$$. If we think about it, we have sufficiently proven what "close" means. Let us put this definition[see footnote 4] up front:

For any $$0< \left | f(x) - L \right\vert <n_{f}$$, there exists a $$0< \left| x_{1} - x \right\vert <n$$.

Now, as mentioned, mathematicians like to "work smarter, not harder," so a mathematician does not want to write out "as $$x$$ approaches $$2$$, $$f(x)$$ approaches $$4$$." Instead, we write out using our fancy notation: $$\lim_{x \to 2}f(x)=4$$. This $$\lim$$ is called "limit." All this notation says is that "as we limit our values of $$x$$ approaching ($$\to$$) $$2$$, we find that $$f(x)$$ approaches $$4$$." Now, you have a sufficient foundation for limits and are ready to begin the proof.


text:
User talk:Ldhank

Please don't remove interwikis. The Wikibooks community discussed their presence in Wikidata and specifically rejected the proposal to remove them locally. Editors here find them valuable. QuiteUnusual (discuss • contribs) 10:31, 19 August 2019 (UTC)
 * ah ok. i did not know that. Five out of seven interwiki's of Waterzooi were dead links. Therfore i deleted them. Ldhank (discuss • contribs) 10:37, 19 August 2019 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Ldhank


text:
User talk:Mmpeet

Hello, your book was created in the wrong namespace and has been moved to LMIs in Control. QuiteUnusual (discuss • contribs) 10:33, 19 August 2019 (UTC)

title: User talk:Mmpeet


text: Category:Book:History of video games

title: Category:Book:History of video games


text:
Organic Chemistry/Haloalkanes/Alkanes

Example

Chloro methane to chloro ethane

CH3Cl +2Na + CH3Cl in presence to dry ether gives CH3-CH3 this reaction is also knows as Wurth's reaction.

It's not the full answer

title: Organic Chemistry/Haloalkanes/Alkanes


text: User:AlexMacArthur

title: User:AlexMacArthur


text:
User talk:Sgdunn9

What is the background to this research?

 * a very brief introduction to what the paper is about,
 * how the research fills a gap in existing knowledge,
 * and why it is important

Where is the research from?

 * Is the location, or group, in which this research was conducted important to share?

 * Do the authors of the research have a track-record/reputation in the field?

 * Are there organisational/sponsorship links that may be a conflict of interest/bias the findings?

What kind of research was this?

 * What kind of study was it? Observational, an RCT, a meta-analysis?
 * Does the level of evidence from these, and other types of studies, differ?

What did the research involve?

 * Provide a lay summary of what the participants did in the study?
 * Was the methodology the best approach?
 * What limitations exist related to the important measures taken, or other aspects of the methodology (participant recruitment for instance)?

What were the basic results?

 * What were the important findings?
 * How did the researchers interpret the results?
 * Do they over-emphasis the implications of their findings?

What conclusions can we take from this research?

 * Provide your own insights on the conclusion (it may not quite be the same as the authors)
 * How do the findings align with other research in the area (in particular more recent publications that won't be mentioned in the paper)

Practical advice

 * What real-world implications does this research have? Examples might includes:
 * criteria for not exercising,
 * recommendations on what exercises to do, how to do them,
 * recommendations on monitoring and progressing exercises within specific populations,
 * considerations for encouraging physical activity within a specific setting, etc
 * Are there other considerations readers should know about before taking on this practical advice? Perhaps health/safety more information/resources?

Further information/resources

 * What further reading may interested readers benefit from?
 * What website, or online organisations, offer further information/support to groups that are linked to your assignment?
 * Consider the audience you are pitching this information for - are the links appropriate (e.g. links to research papers are probably not appropriate if the information is for patients, but it may be if it is for other allied health professionals)

What is the background to this research?

 * a very brief introduction to what the paper is about,
 * how the research fills a gap in existing knowledge,
 * and why it is important

Where is the research from?

 * Is the location, or group, in which this research was conducted important to share?

 * Do the authors of the research have a track-record/reputation in the field?

 * Are there organisational/sponsorship links that may be a conflict of interest/bias the findings?

What kind of research was this?

 * What kind of study was it? Observational, an RCT, a meta-analysis?
 * Does the level of evidence from these, and other types of studies, differ?

What did the research involve?

 * Provide a lay summary of what the participants did in the study?
 * Was the methodology the best approach?
 * What limitations exist related to the important measures taken, or other aspects of the methodology (participant recruitment for instance)?

What were the basic results?

 * What were the important findings?
 * How did the researchers interpret the results?
 * Do they over-emphasis the implications of their findings?

What conclusions can we take from this research?

 * Provide your own insights on the conclusion (it may not quite be the same as the authors)
 * How do the findings align with other research in the area (in particular more recent publications that won't be mentioned in the paper)

Practical advice

 * What real-world implications does this research have? Examples might includes:
 * criteria for not exercising,
 * recommendations on what exercises to do, how to do them,
 * recommendations on monitoring and progressing exercises within specific populations,
 * considerations for encouraging physical activity within a specific setting, etc
 * Are there other considerations readers should know about before taking on this practical advice? Perhaps health/safety more information/resources?

Further information/resources

 * What further reading may interested readers benefit from?
 * What website, or online organisations, offer further information/support to groups that are linked to your assignment?
 * Consider the audience you are pitching this information for - are the links appropriate (e.g. links to research papers are probably not appropriate if the information is for patients, but it may be if it is for other allied health professionals)

title: User talk:Sgdunn9


text:
User:MattG98/sandbox

This article is a critical analysis of the research article "Both aerobic Endurance and strength training programmes improve cardiovasular health in obese adults" by Inga E. Schjerve et al.

What is the background to this research?

 * The aim of this study was to look at the efficencey of moderate and high intensity level aerobic training and strength training on improving cardiovascular health in obsese and overweight populations.
 * and why it is important
 * and why it is important

Where is the research from?

 * Is the location, or group, in which this research was conducted important to share?

 * Do the authors of the research have a track-record/reputation in the field?

 * Are there organisational/sponsorship links that may be a conflict of interest/bias the findings?

What kind of research was this?

 * What kind of study was it? Observational, an RCT, a meta-analysis?
 * Does the level of evidence from these, and other types of studies, differ?

What did the research involve?

 * Provide a lay summary of what the participants did in the study?
 * Was the methodology the best approach?
 * What limitations exist related to the important measures taken, or other aspects of the methodology (participant recruitment for instance)?

What were the basic results?

 * What were the important findings?
 * How did the researchers interpret the results?
 * Do they over-emphasis the implications of their findings?

What conclusions can we take from this research?

 * Provide your own insights on the conclusion (it may not quite be the same as the authors)
 * How do the findings align with other research in the area (in particular more recent publications that won't be mentioned in the paper)

Practical advice

 * What real-world implications does this research have? Examples might includes:
 * criteria for not exercising,
 * recommendations on what exercises to do, how to do them,
 * recommendations on monitoring and progressing exercises within specific populations,
 * considerations for encouraging physical activity within a specific setting, etc
 * Are there other considerations readers should know about before taking on this practical advice? Perhaps health/safety more information/resources?

Further information/resources

 * What further reading may interested readers benefit from?
 * What website, or online organisations, offer further information/support to groups that are linked to your assignment?
 * Consider the audience you are pitching this information for - are the links appropriate (e.g. links to research papers are probably not appropriate if the information is for patients, but it may be if it is for other allied health professionals)

title: User:MattG98/sandbox


text:
User:J.garbuio/sandbox

The research critique is about the study of 'Sedentary time in U.S. older adults associated with disability in activities of daily living independent of physical activity' by Dunlop, Song, Arntson, Semanik, Lee, Chang and Hootman.

 Measuring physical activity in obese prolonged sedentary older adults

What is the background to this research?
Physical inactivity and sedentary behaviors causing obesity and leave of absences in jobs. Approximately 40% of individuals around the world with chronic diseases were not meeting a minimum of 150 minutes of moderate intensity per week of physical activity and the percentage grew higher as the countries with high income raised also. As of not only meeting the recommended guidelines for physical activity, in 2014, this would see 30% of the world's population being obese or overweight.

The purpose of the study that makes this important is to get workers that are prolonged, sedentary throughout their waking hours and getting adults more physical activate throughout the day while their awake to decrease the potential risk of increasing the severity of obesity in adults where they're are sedentary for long periods of their waking day. In 2014, the UK’s Office for National Statistics emphasized that approximately 12% of the 131 million working days lost were from either stress, anxiety and/or depression which could of lead from how they see themselves or others see them. The research on the adult population with desk or screen watching jobs are getting more obese and less activate.

Where is the research from?
Dunlop, Song, Arntson, Semanik, Lee, Chang and Hootman conducted the research in the United States of America by the National Health and Nutrition Examinations Surveys (NHANES). This research was conducted using cross-sectional study to see how the effect on health outcomes improve overtime by introducing less sedentary time (<1.5 metabolic equivalents). Even though the research was conducted in the USA, the findings are still applicable for Australia.

What did the research involve?
The participation of 2,286 adults, consisting of 1,159 males and 1,127 females aged 60 years or older, were being measured for seven consecutive days to measure their sedentary time and their amount of moderate-vigorous time throughout those seven days.

The NHANES collected and checked the data by an accelerometer monitor that the NHANES placed on the participants right hip. The accelerometer monitor was to be worn by the participants while they were awake and not doing any water activities such as showers or submerging the monitor underwater that would disturb or ruin the results in the monitor. The accelerometer monitor was placed on the right hip to allow the participants in the study to be able to do their regular daily living lifestyle with only a minimal disturbance to the participants.

Participants were excluded from the research if they did not have a valid amount of time of wearing the monitor of 10 or more hours throughout each day and also the participants were excluded if there was not four or more of the seven days not being valid of wearing the monitor 10 or more hours per day. The study with the accelerometer monitors on the participants was to observe the average time in hours and percentage each participant spent sedentary during the time the participants were wearing the monitor.

What were the basic results?
After the seven day period and when all the monitors were collected and analysed, the results found that out of the 2,286 participants aged 60 years and older, the average time the monitor was worn was 14 hours per day. Of that 14 hours, 63.4% of the participants registered to be sedentary for at least 9 or more hours per day.

The researchers for the study found that the participants that were not meeting the physical activity guidelines and present a chronic illness was relevant to those participants having a higher sedentary behavior.

What conclusions can we take from this research?
The study that the researchers conducted was a valuable design that could continue to be used and/or add modifications to the study to decrease the amount of time older adults spend sedentary and in type to sedentary position such as lying at home or sitting at work and increase physical activity among the older adult population.

Interventions were implemented to improve health outcomes in older adults proving programs that encourage cutting sedentary behaviors by involving more moderate-vigorous activities that will affect the future public health positively.

Practical advice
Further research on a longer period of study to allow variance in the participants week to week schedules. Allowing devices that can measure upper body movements as the monitor was placed on the hip so it limits to detect upper body movements as well as activities such as water activities were not able to be measured if the participants went for a swim or aqua aerobics which could alter the results. Participants wearing the monitor could stimulate them to participate with others or themselves to overcome certain psychological obstacles and make the participates engage in physical activity to make themselves seem health to the researchers eyes.

Further information/resources
Environmental Determinants of Physical Activity and Sedentary Behavior

Appropriate Physical Activity Intervention Strategies for Weight Loss and Prevention of Weight Regain for Adults

Objectively Measured Sedentary Time, Physical Activity, and Metabolic Risk

Reducing Sedentary Behavior: Role in Modifying Physical Activity

Television Viewing Time is Associated with Overweight/Obesity Among Older Adults, Independent of Meeting Physical Activity and Health Guidelines

Physical activity of Canadian adults: Accelerometer results from the 2007 to 2009 Canadian Health Measures Survey

title: User:J.garbuio/sandbox


text:
User:Breecav/sandbox

This article is a critical analyses of the paper "Relation of Cardiovascular Fitness and Physical Activity to Cardiovascular Disease Risk Factors in Children and Adults" (James S. Fallis, 1998)

What is the background to this research?

 * a very brief introduction to what the paper is about,
 * how the research fills a gap in existing knowledge,
 * and why it is important

Where is the research from?

 * Is the location, or group, in which this research was conducted important to share?

 * Do the authors of the research have a track-record/reputation in the field?

 * Are there organisational/sponsorship links that may be a conflict of interest/bias the findings?

What kind of research was this?

 * What kind of study was it? Observational, an RCT, a meta-analysis?
 * Does the level of evidence from these, and other types of studies, differ?

What did the research involve?

 * Provide a lay summary of what the participants did in the study?
 * Was the methodology the best approach?
 * What limitations exist related to the important measures taken, or other aspects of the methodology (participant recruitment for instance)?

What were the basic results?

 * What were the important findings?
 * How did the researchers interpret the results?
 * Do they over-emphasis the implications of their findings?

What conclusions can we take from this research?

 * Provide your own insights on the conclusion (it may not quite be the same as the authors)
 * How do the findings align with other research in the area (in particular more recent publications that won't be mentioned in the paper)

Practical advice

 * What real-world implications does this research have? Examples might includes:
 * criteria for not exercising,
 * recommendations on what exercises to do, how to do them,
 * recommendations on monitoring and progressing exercises within specific populations,
 * considerations for encouraging physical activity within a specific setting, etc
 * Are there other considerations readers should know about before taking on this practical advice? Perhaps health/safety more information/resources?

Further information/resources

 * What further reading may interested readers benefit from?
 * What website, or online organisations, offer further information/support to groups that are linked to your assignment?
 * Consider the audience you are pitching this information for - are the links appropriate (e.g. links to research papers are probably not appropriate if the information is for patients, but it may be if it is for other allied health professionals)

title: User:Breecav/sandbox


text:
User talk:ADIG1212

Hello, my name is Anthony. I like dogs.

Background
Ex- Army- Airborne battle group.

Interests
Dogs

Wikibooks Projects
Nil at the moment

title: User talk:ADIG1212


text:
User:LachlanC2000/sandbox

This is a critique of the research article: Papaioannou A, Adachi J, Winegard K, Ferko N, Parkinson W, Cook R, et al. Efficacy of home-based exercise for improving quality of life among elderly women with symptomatic osteoporosis-related vertebral fractures. Osteoporosis International. 2003;14(8):677-82.

It was written for an assignment in the unit, Health, Disease and Exercise at the University of Canberra.

What is the background to this research?
Osteoporosis is defined as a systemic skeletal disease characterised by low bone mass and microarchitectural deterioation of bone tissue, which leads to increased bone fragility and higher risk of fracture.

Research into osteoporosis and how the severity of the condition can be alleviated or managed via the use of exercise is crucial as osteoporosis can occur in many populations and ethnic groups, with elderly people being at the greatest risk. Hence, the possible benefits that physical activity have on this disease need to be explored to help those suffering with the disease and possibly find new alternatives to be used in the future.

Where is the research from?
This research article was published in International Osteoporosis Journal.

The primary author of this article is Dr. Alexandra Papaioannou. She is a geriatrician and has a special interest in osteoporosis, pharmacology and falls and quality of life in the elderly population. She has been involved with many papers regarding osteoporosis. In addition, many of the other authors have completed research on osteoporosis in some capacity.

What kind of research was this?
This study was a randomized control trial. One group participated in the intervention and the control group did not participate in the intervention but were used for comparison later.

Randomized control trials are considered to be the most credible method of finding out whether a cause and effect relationship exists.

What did the research involve?
The study involved 74 female participants over the age of 60 split randomly into either an exercise or control group.

All subjects had to meet the following criteria:
 * Lumbar spine bone mineral density ≥ -2.5 SD below young adult mean
 * At least one vertebral fracture

In this study the exercise group had to complete exercises for 60 minutes a day, 3 days a week. Their exercises included stretching, upper and lower body resistance training and aerobics. The control group had no guidelines for exercise and were told to continue with their usual daily activities.

At baseline, 6 months and 12 months, participants had to complete the Osteoporosis Quality of Life Questionnaire (OQLQ), the Sickness Impact Profile (SIP), the sway test and the timed up and go test and at only baseline and 12 months participants had to undergo a dual-energy X-ray absorptiometry (DEXA) scan assessing bone mineral density at the lumbar spine and femoral neck.

A limitation of the study was that the OQLQ had not been validated, therefore the results might not actually be of any significance. Also, some of the participants had asymptomatic vertebral fractures, but it was thought that the intervention would be best suited for those with symptomatic fractures. Furthermore, 17 participants failed to complete the study and adherence to the intervention dropped after the first 6 months, this may be attributed to the fact that there was decreased supervision by the therapist.

What were the basic results?
6 Months
 * Exercise group had greater improvement in symptoms (P=0.03) specifically in lack of energy, pain from carrying, pain from standing, pain from walking and tiredness.
 * Change in emotion was displayed (P=0.01) specifically they were not as upset about having the disease.
 * Change in leisure/social (P=0.03) specifically being able to do more physical activity and being able to go on holidays.
 * In the sway test, participants had improved in range of displacement (P=0.01)

12 Months
 * Improvements were shown in symptoms (P=0.01) such as pain from bending, standing and walking were all maintained.
 * There was improvements in activities of daily living (P=0.04)
 * In the sway test there was significant improvements for results in lateral and anteroposterior directions as well as velocity of movement (P=0.01).

The results are consistent with what the researchers concluded. The researchers deducted that over a 6 month period, a home exercise program improved quality of life in symptoms, emotion, leisure and social activities. Furthermore they believed that if home based exercises are completed over 12 months, improvements recorded at 6 months will be sustained.

What conclusions can we take from this research?
Despite not having the largest final group size, this article has put forward evidence that home-based exercises can improve quality of life in post-menopausal women with osteoporosis related vertebral fractures. However, the validity of this evidence is negatively impacted due to not using a validated test (OQLQ) and low adherence to the intervention. In saying that the data provided is valuable in showing how quality of life can be improved in this population. The findings in this paper correlate with other research done in this area, in that exercise induces positive effects in women with osteoporosis whether or not they have a vertebral fracture. Nevertheless, more research should be conducted on what exercises prove to be more useful in improving quality of life and overall health of those suffering with osteoporosis and/or vertebral fractures.
 * Provide your own insights on the conclusion (it may not quite be the same as the authors)
 * How do the findings align with other research in the area (in particular more recent publications that won't be mentioned in the paper)

Practical advice

 * What real-world implications does this research have? Examples might includes:
 * criteria for not exercising,
 * recommendations on what exercises to do, how to do them,
 * recommendations on monitoring and progressing exercises within specific populations,
 * considerations for encouraging physical activity within a specific setting, etc
 * Are there other considerations readers should know about before taking on this practical advice? Perhaps health/safety more information/resources?

Further information/resources

 * What further reading may interested readers benefit from?
 * What website, or online organisations, offer further information/support to groups that are linked to your assignment?
 * Consider the audience you are pitching this information for - are the links appropriate (e.g. links to research papers are probably not appropriate if the information is for patients, but it may be if it is for other allied health professionals)

title: User:LachlanC2000/sandbox


text:
User:A.scott8689/sandbox

Benavides, S. & Caballera, J. Ashtanga yoga for children and adolescents for weight management and psychological well being: An uncontrolled open pilot study. Compl Therapies in Clinical Practice. Edinburg, The University of Texas. 2009: 15;110-114.

What is the background to this research?

 * a very brief introduction to what the paper is about,
 * how the research fills a gap in existing knowledge,
 * and why it is important

Where is the research from?

 * Is the location, or group, in which this research was conducted important to share?

 * Do the authors of the research have a track-record/reputation in the field?

 * Are there organisational/sponsorship links that may be a conflict of interest/bias the findings?

What kind of research was this?

 * What kind of study was it? Observational, an RCT, a meta-analysis?
 * Does the level of evidence from these, and other types of studies, differ?

What did the research involve?

 * Provide a lay summary of what the participants did in the study?
 * Was the methodology the best approach?
 * What limitations exist related to the important measures taken, or other aspects of the methodology (participant recruitment for instance)?

What were the basic results?

 * What were the important findings?
 * How did the researchers interpret the results?
 * Do they over-emphasis the implications of their findings?

What conclusions can we take from this research?

 * Provide your own insights on the conclusion (it may not quite be the same as the authors)
 * How do the findings align with other research in the area (in particular more recent publications that won't be mentioned in the paper)

Practical advice

 * What real-world implications does this research have? Examples might includes:
 * criteria for not exercising,
 * recommendations on what exercises to do, how to do them,
 * recommendations on monitoring and progressing exercises within specific populations,
 * considerations for encouraging physical activity within a specific setting, etc
 * Are there other considerations readers should know about before taking on this practical advice? Perhaps health/safety more information/resources?

Further information/resources

 * What further reading may interested readers benefit from?
 * What website, or online organisations, offer further information/support to groups that are linked to your assignment?
 * Consider the audience you are pitching this information for - are the links appropriate (e.g. links to research papers are probably not appropriate if the information is for patients, but it may be if it is for other allied health professionals)

title: User:A.scott8689/sandbox


text:
Exercise as it relates to Disease/High Intensity Interval Exercise and Colon Cancer

The following is a critque of the journal article "acute high intensity interval exercise reduces colon cancer cell growth " written by James L. Devin, Michelle M. Hill, Marina Mourtzakis, Joe Quadrilatero, David G. Jenkins and Tina L. Skinner.

What is the background to this research?
According to the World Cancer Research Fund, colorectal cancer is the third most common form of cancer in men and second most common in women. Exercise is known to have significant preventative effect on the prevalance of cancer. Exercise also decreases mortality rates in colorectal cancer survivors. A suggested mechanism for these benefits of exercise has to do with the release of anti-inflammatory chemicals, called cytokines, after exercise. The study by Devin et al. aims to observe the immediate and long term effects of these cytokines on colorectal cancer cell growth after high intensity interval exercise (HIIE). HIIE was chosen because it causes a larger release of these cytokines than moderate intensity exercise. The results of this study will help health professionals to better understand one of the possible mechanisms behind the benefits of exercise as it relates to colorectal cancer and inform a more accurate prescription of exercise to colorectal cancer survivors. This research also fills a gap in current knowledge in regards to the acute and chronic impact exercise has on colorectal cancer growth.

Where is the research from?
This study comes from The University of Queensland in Brisbane, Australia. It was funded by The University of Queensland's HABS/MABS Collaboration Seeding Grant, as well as their Graduate School International Travel Award. Additional funding for this project came from the Sports Medicine and Research Foundation. The research was published in The Journal of Physiology, a long running, reputable journal. Due to the majority of the funding coming from The University of Queensland, and not a major company, there is reason to assume that no bias is being caused by the financial bodies behind the study.

What kind of research was this?
This study consisted of 2 kinds of research: It is important to note that "in vitro" studies are not always reflective of all the physiological mechanisms occurring at the cellular level due to the separation of the cells from the whole organism. Since the cells are looked at in isolation, other reactions or processes that may affect the cells within the body cannot be observed.
 * Human Clinical Trial
 * In Vitro

Human Clinical Trial
A group (20 people) of male, colorectal cancer survivors, at least one month into remission, were split into 2 groups of 10. The participants were all older than 40 years of age however the average age was 66. It is known that as aging occurs, there is a decreased cytokine response to exercise. Due to this impaired cytokine response to exercise, the effects on cancer cells may be different when exercising younger patients. Group 1 (acute) completed a single exercise session of 4 sets of 4-minute work intervals at 85-95% of their maximum heart rate. 3 minutes of active recovery was given between sets and a 10 minute warm up was completed prior to the first work interval commencing. The exercise was completed on an exercise bike. Cytokine response to exercise is typically lower in cycling when compared to running, however, cycling is likely to be more appropriate for this population due to the age of the participants and the higher impact nature of running. Group 2 (short term) completed the same exercise session 3 times a week for 4 weeks. In the acute treatment group, blood was sampled prior to exercise (fasted), immediately after exercise and 120 minutes post exercise. The short term treatment group had blood samples taken 3-7 days prior to beginning the intervention and following the intervention. Blood samples in this group were taken at the same time of day, this is an important control measure as cytokine secretion from muscles, when not prompted by exercise, follow a circadian rhythm. Blood samples were centrifuged to create serum and were analysed for both groups to determine levels of the cytokines: interleukin-6 (IL-6), interleukin-8 (IL-8) and tumour necrosis factor alpha (TNF-a).

In Vitro Study
Two genetically different colorectal cancer cell lines were obtained. The genetic differences in cells were chosen as different genes produce different types of mutation in colorectal cancer cells. By doing this, the research from this leg of the study may have broader implications on differing mutations of colorectal cancer. Cells were then incubated in serum from pre exercise, immediately after, or 120 minutes post exercise. Serum used was obtained from the blood samples of the acute treatment group. Incubation lasted for 72 hours with measures of cell number and cell death collected at 24, 48 and 72 hours.

What were the basic results?

 * All cytokines were significantly elevated from rest levels immediately after exercise but returned to resting levels 120 minutes after exercise.
 * No change in resting levels of the cytokines measured was observed after a 4 week HIIE training intervention.
 * Both genetic strains of colorectal cancer cells had a significantly decreased cell count at 24, 48 and 72 hours when incubated in serum from immediately after exercise when compared to serum collected at rest.
 * When incubated in serum from 120 minutes post exercise, neither genetic cancer cell variation decreased in cell count over 72 hours when compared to resting serum incubation.

What conclusions can we take from this research?
HIIE causes elevations in cytokines which return to resting levels after 120 minutes. No long term changes in the resting levels of these cytokines are caused by this style of exercise. Due to the isolated nature of the in vitro study, which the author acknowledges, it cannot be said that these elevations of cytokines in the body would have the same effect or magnitude of effect on colorectal cancer cells as seen in the in vitro arm of the study. However, when exposed to serum with elevated levels of these cytokines, colorectal cancers experience decreases in cell count over 72 hours. It is important to note that the cell count decrease was seen when cancer cells were exposed to a consistent level of these cytokines for at least 24 hours, whereas in the body the cytokine levels were only seen to remain elevated for less than 2 hours. This significantly decreases the window of time the cytokines have to induce cell death and highlights the need for research into the in body effect high intensity interval exercise has on colorectal cancer cells. The author also brings to light other limitations to the study such as the lack of a non-exercising control and the small sample size used. The author appropriately suggests that further research could also be conducted into the changes in the cytokine response to exercise over a multiple week period of HIIE training.

Practical advice
While the body is a complex integrated system and colorectal cancer is a multifactorial disease, exercise is known to decrease mortality in survivors of the disease. The cytokine effect on cell decreases in colorectal cancer could be one of many mechanisms causing this outcome and may even have implications for prevention of the disease as well. For this reason engaging in an exercise program involving high intensity interval training for colorectal cancer survivors and the general population may be of great benefit. It is critical not to forget all the other health benefits of exercise not focussed on by this study. Always consult a doctor before commencing an exercise program, especially if you wish to engage in high intensity exercise.

 * 1) Reduced Risk: Research suggests that regular physical activity, including HIIE, is associated with a decreased risk of colon cancer. A meta-analysis published in the British Journal of Cancer in 2016 found that higher levels of physical activity were associated with a 24% lower risk of colon cancer compared to individuals with low levels of physical activity.
 * 2) Mechanisms: HIIE and other forms of exercise may influence multiple biological mechanisms that could reduce the risk of colon cancer. These mechanisms include decreased inflammation, improved insulin sensitivity, regulation of hormone levels (such as insulin and insulin-like growth factor-1), and enhanced immune function. These effects may collectively contribute to a lower risk of developing colon cancer.
 * 3) Body Weight and Fat Reduction: HIIE can be particularly effective in reducing body weight and fat mass. Obesity and excess body fat are established risk factors for colon cancer. By helping individuals achieve and maintain a healthy weight, HIIE may indirectly lower the risk of colon cancer.
 * 4) Improved Gut Health: Exercise, including HIIE, may positively influence gut health by promoting a diverse and healthy gut microbiota composition. A balanced gut microbiota has been linked to a reduced risk of various diseases, including colon cancer.
 * 5) Enhanced Detoxification: Regular exercise, including HIIE, can enhance detoxification processes within the body, potentially reducing the accumulation of harmful substances in the colon that could contribute to carcinogenesis.
 * 6) Adherence to Guidelines: Current guidelines for cancer prevention typically recommend regular physical activity as part of a healthy lifestyle. HIIE can offer a time-efficient way to meet these guidelines, as it involves short bursts of intense exercise interspersed with brief periods of rest.
 * 7) HIIE can be particularly effective in reducing body weight and fat mass

=== Further information/resources === The following websites provide information on the signs, symptoms, diagnosis and treatment for colorectal cancer.

 * Cancer Council Australia - Bowel/Colorectal Cancer

 * American Cancer Society - What is Colorectal Cancer?


text: User:Tammypovey/sandbox

title: User:Tammypovey/sandbox


text:
User:U3175233/sandbox

The following article is a critique of the research paper 'A randomised controlled trial of a home based exercise programme to reduce the risk of falling among people with Parkinson's disease', published in the Journal of Neurology, Neurosurgery, and Psychiatry, 2007.

What is the background to this research?
Parkinson's Disease (PD) is a long-term degenerative disorder which effects the brain and nervous system. Main symptoms include shaking of the arm or hand, muscle rigidity, slowed gait movement, and postural instability. PD affects around 1 in 1000 individual's over 60 years worldwide, and approximately 81,000 individuals in Australia.

Frequent falls in older adults can have severe consequences, including hip fractures and increased weakness of bones, which decreases quality of life. Falling is the second most common cause for hospitalisation in individuals with PD. 51.9% of all falls resulting in hospitalisation, occurred within the home. Therefore, the effect of home-based exercises on decreasing falls, by increasing muscle strength and reducing the fear of falling, should be of concern. Research into the topic can assist in reducing the economic burden of hospitalisations in PD individuals due to frequent falls.

Where is the research from?
This research paper was published by the Journal of Neurology, Neurosurgery, and Psychiatry, a well reputed medical-related journal accessible worldwide.

The primary author, Ann Ashburn, has been involved in the production of around 100 research papers. Ashburn's research focuses on physical recovery after sudden medical events, such as a stroke, or long-term disorders, such as PD.

The study took place in Dorest, UK, with participants chosen from three PD specialists within the area. Average age of the participants was 72 years ± 9.2, and time since PD diagnosis ranged from 1 to 31 years. Similar studies have been conducted in Australia, providing a level of relevance to Australian individuals with PD.

What kind of research was this?
This research was a randomised controlled trial (RCT) which is considered the gold standard in research. RCT aims at limiting any discrepancies between the intervention and control groups. Progress made from the intervention is highlighted when compared to the control group.

One limitation of the study was self-reporting. Individuals could have inaccurately defined or underestimated falls and near falls that occurred. Therefore, the exercise intervention may have made a larger impact on the patients than the study suggests.

What did the research involve?
The research involved 142 participants diagnosed with varying severity of PD, who were randomised into an intervention group (n=70) and a control group (n=72). Eligibility criteria required individuals to be living at home, independently mobile, and experienced more than one fall 12 months prior to the study. The number of participants declined due to dislike of exercise, hospitalisation, or passed away. The participants were required to self-report any fall and near fall events, and injuries that may have occurred as a result. 134 self-reporting diaries were available at completion of the study (n = 67 from each group).

The study defines a fall as "an event that resulted in a person coming to rest unintentionally on the ground or other lower level, not as a result of a major intrinsic event", and a near fall as "an occasion on which and individual felt that they were going to fall but did not actually do so".

The exercise completed by the intervention group involved muscle strengthening, range of movement, balance training, and walking, which promoted the prevention of falls, initiation of movement, and strength compensation. Exercises were taught by a physiotherapist. After 6 weeks, the intervention group were contacted to discuss any arising problems with the prescribed exercises. Follow ups were then completed after 8 weeks and 6 months.

The control group received their usual care.

What were the basic results?
A decrease in falls rate and repeat falling for the intervention group (see Table 1), as well as a decreased rate of injuries requiring medical attention (see Table 2) resulted from the study. Although improvements in the individual's quality of life occurred, these particular results did not reach statistical significance. As time progressed, the study revealed a declining pattern in quality of life for the control group, with greater rates of falls (see Table 1) and subsequent related injuries (see Table 2).

Lower rates of near falls were recorded in the intervention group in comparison to the control. This reached statistical significance.

Table 1: Rate of falling, repeated falling, and near falling in participants at 8 week and 6 months follow ups.

Table 2: Number of Fall Related Injuries occurring over the course of the study requiring medical attention.

What conclusions can we take from this research?
As this study was relatively small (n = 142), only representing 0.09% of the PD population within the UK (estimated 145,000 cases recorded in 2018), the evidence of using exercise as a disease intervention in individuals with PD can be taken into consideration when prescribing treatment plans. However, interventions used in this research cannot be recommended for the whole population as each individual is different.

The research findings suggest that near fall preventions become effective after implementing an exercise intervention program. Number of fall related injuries also decrease. As a result, quality of life and overall health can increase.

The study shows that those with lower disease severity and lower fall rate benefited greatly from the intervention. Therefore, it is suggested that an exercise program should be prescribed shortly after diagnosis, to decrease the risk of falls, as disease severity worsens.

Practical advice
Even though there is currently no known cure, there are a number of different ways PD symptoms can be treated. Possible treatments include medication (increases dopamine activity in the brain), surgery (reduces the tremor or shakiness in limbs), and exercise (improves balance, stability and movement). General exercise recommendations for individuals with PD are explained in Table 3, though the prescription of each treatment is dependent on disease severity and should be prescribed by an accredited health professional.

Individuals with PD should remain active while still maintaining skills required for everyday living. Assistance may be required in performing these skills; however, it is critical for the individual to remain as independent as possible to improve their quality of life.

Table 3: ACSM's General Exercise Recommendations for Individuals with Parkinson's Disease.

Further information/resources
Below are a list of websites that further explain PD and ways it can be managed.

It is worth noting that these websites do not substitute for medical advice. It is best to consult with a medical professional to manage and prescribe PD treatments.

 * Exercise for People with Parkinson's

 * Living with Parkinson's Information Sheets
 * Parkinson's Disease Information

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