# Tagged Questions

linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics

3answers
479 views

### Why everytime the final number comes the same?

I have come across an interesting puzzle. Write $20$ numbers. Erase any two number say $x$ and $y$ and and replace with $\text{Number}_{new} = xy/(x + y)$ OR $\text{Number}_{new}= x + y + xy$ ...
5answers
1k views

### No radical in the denominator — why?

Why do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to ...
2answers
574 views

### Is this a known algebraic identity?

In the course of analyzing a certain Markov chain, I once had to prove the following algebraic identity. Is there a slick or known proof? For $n$-tuples $(x_1,x_2,\dots, x_n)$ of positive real ...
11answers
5k views

### Derivation of the formula for the vertex of a Parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $y = a x^2 + b x + c$ My teacher gave me the formula: $x = -\frac{b}{2a}$ as the $x$ ...
2answers
304 views

### How can I compute $\sum\limits_{k = 1}^n \frac{1} {k + 1}\binom{n}{k}$?

This sum is difficult. How can I compute it, without using calculus? $$\sum_{k = 1}^n \frac1{k + 1}\binom{n}{k}$$ If someone can explain some technique to do it, I'd appreciate it. Or advice using ...
6answers
332 views

### The existence of partial fraction decompositions

I'm sure you are all familiar with partial fraction decomposition, but I seem to be having trouble understanding the way it works. If we have a fraction f(x)/[g(x)h(x)], it seems only logical that it ...
5answers
283 views

### How to “Re-write completing the square”: $x^2+x+1$

The exercise asks to "Re-write completing the square": $$x^2+x+1$$ The answer is: $$(x+\frac{1}{2})^2+\frac{3}{4}$$ I don't even understand what it means with "Re-write completing the square".. ...
3answers
645 views

### How to find the sum of the following series

How can I find the sum of the following series? $$\sum_{n=0}^{+\infty}\frac{n^2}{2^n}$$ I know that it converges, and Wolfram Alpha tells me that its sum is 6 . Which technique should I use to ...
4answers
402 views

### Steps to solve  this system of equations

I want to solve this system of equations, I have been out of Maths for a long !! $$\sqrt{x} + y = 7$$ $$\sqrt{y} + x =11.$$ Just wondering easiest step to find values for $x$ and $y$ from the above ...
1answer
76 views

### What is the minimum number of blocks to build this?

A rectangular solid is built using $N$ cubes of a side length of 1cm. When viewed from such an angle such that only 3 of the sides of the rectangular solid are visible, there are 231 $cm^2$ of area ...
4answers
274 views

### Algorithms for “solving” $\sqrt{2}$

The very first words out of my mouth need to be this... "Solving" is the wrong term since I am speaking about irrational numbers. I just don't know which word is the correct word... So that can be ...
4answers
818 views

### Geometric mean never exceeds arithmetic mean

This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.) The geometric ...
5answers
1k views

### How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$?

I'm doing exercise on discrete mathematics and I'm stuck with question: If $f:Y\to Z$ is an invertible function, and $g:X\to Y$ is an invertible function, then the inverse of the composition \$(f ...
2answers
217 views

### If $2 x^4 + x^3 - 11 x^2 + x + 2 = 0$, then find the value of $x + \frac{1}{x}$?

If $2 x^4 + x^3 - 11 x^2 + x + 2 = 0$, then find the value of $x + \frac{1}{x}$ ? I would be very grateful if somebody show me how to factor this polynomial by hand, as of now I have used to ...
3answers
276 views

### Question about solving absolute values

I solved the following problem from by book, but the answer of this problem at the end of book is x < = 3 Please tell me how can i get this answer.
3answers
209 views

### Finding the error in a proof

I have a "proof" that has an error in it and my goal is to figure out what this error is. The proof: If x = y, then  \begin{eqnarray} x^2 &=& xy \nonumber \\ x^2 - y^2 &=& xy - ...
4answers
445 views

### Solving a quadratic inequality

I am solving the following inequality, please look at it and tell me am i write or not because this is an example in Howard Anton's book, i solved it on my own as given below but in the book it is ...
2answers
94 views

### How do I find the delta analytically for $f(x)$ with a degree other than $1$

So I know how to find the $\delta$ in $f(x)$ that can be factored into a degree of $1$, or I can solve it by solving $L + \epsilon = f(x)$ then finding the distance from $a$. But how do I find the ...
4answers
269 views

### Power summation of $n^3$ or higher [duplicate]

Possible Duplicate: why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$ If I want to find the formula for $$\sum_{k=1}^n k^2$$ I would do the next: (n+1)^2 = a(n+1)^3 ...
1answer
163 views

### Maximum inequality

Can anyone show that ...
4answers
177 views

### the sum of powers of $2$ between $2^0$ and $2^n$

Lately, I was wondering if there exists a closed expression for $2^0+2^1+\cdots+2^n$ for any $n$?
2answers
82 views

### Problem with Abel summation

Let $$S_n=\sum_{i=1}^{n}\sin k,\quad S_0=0.$$ Then $$\sum_{k=1}^{n}\frac{\sin k}{k}=\sum_{k=1}^{n}\frac{S_k-S_{k-1}}{k}=\frac{S_n}{n}+\sum_{k=1}^{n-1}\frac{S_k}{k(k+1)}.$$ Could someone please ...
2answers
230 views

### How to find all real solution to satisfy this equation without casework or bruteforce?

How to find all real solution to satisfy this equation without casework or bruteforce? a+b+c=abc Thanks in advance!
16answers
2k views

### Interesting calculus problems of medium difficulty?

I would like this to be a community wiki. But, I don't know how to do that here. I would like to know sources, and examples of good "challenge" problems for students who have studied pre-calculus ...
5answers
403 views

### Solving a peculiar system of equations

I have the following system of equations where the $m$'s are known but $a, b, c, x, y, z$ are unknown. How does one go about solving this system? All the usual linear algebra tricks I know don't apply ...
4answers
392 views

### How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?

Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$ Apart from induction,I tried with Wolfram Alpha to check the validity,but ...
2answers
406 views

### Given $n! = c$, how to find $n$?

I'm dealing with a time-complexity problem in which I know the running time of an algorithm: $$t = 1000 \mathrm{ms} .$$ I also know that the algorithm is upper bounded by $O(n!)$. I want to know ...
7answers
1k views

### Algebraic Identity $a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$

Prove the following: $\displaystyle a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$. So one could use induction on $n$? Could one also use trichotomy or some type of combinatorial ...
4answers
201 views

### Should the domain of a function be inferred?

It is a common practice to have students of elementary algebra infer the domain of a function as an exercise. I believe this is contrary to the spirit of the definition of a function as a collection ...
2answers
166 views

### Determining the number $N$

Let $1 = d_1 < d_2 <\cdots< d_k = N$ be all the divisors of $N$ arranged in increasing order. Given that $N=d_1^2+d_2^2+d_3^2+d_4^2$, determine $N$. The divisors include $N$. It seems that ...
4answers
465 views

### Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$.

Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$ I have no idea how to deal with this crazy question. One idea came into my mine is factorization, but I can't go on... Can anyone help me please? ...
2answers
256 views

### Algebraic equation problem - finding $x$

$$(x^2 +100)^2 =(x^3 -100)^3$$ How to solve it?
2answers
263 views

### Prove the following property of $f(x)$?

Let $$f(x)=|a_1\sin(x)+a_2\sin(2x)+a_3\sin(3x)+...+a_n\sin(nx)|.$$ Given that $f(x)$ is less than or equal to $|\sin(x)|$ for all $x$, prove that $|a_1+a_2+a_3+....|$ is less than or equal to ...
6answers
423 views

### What is the result of $\lim\limits_{x \to 0}(1/x - 1/\sin x)$?

Find the limit: $$\lim_{x \rightarrow 0}\left(\frac1x - \frac1{\sin x}\right)$$ I am not able to find it because I don't know how to prove or disprove $0$ is the answer.
5answers
475 views

### Prove this number fact

Prove that $x \neq 0,y \neq 0 \Rightarrow xy \neq 0$. Suppose $xy = 0$. Then $\frac{xy}{xy} = 1$. Can we say that $\frac{xy}{xy} = 0$ and hence $1 = 0$ which is a contradiction? I thought ...
4answers
109 views

### Working with proofs help?

I'm trying to study for my midterm and doing some random practise questions to work with proofs. However I'm stuck on, as the only way I know how to prove it is through plugging in numbers, however as ...
1answer
243 views

### One sum of squares and two Diophantine equations

This question comes from trying to see why 24 is the only non-trivial value of $n$ for which $$1^2+2^2+3^2+\cdots+n^2$$ is a perfect square. To this end, let $m,n \in \mathbb N$ be such that ...
3answers
492 views

### Gaussian proof for the sum of squares?

There is a famous proof of the Sum of integers, supposedly put forward by Gauss. $$S=\sum\limits_{i=1}^{n}i=1+2+3+\cdots+(n-2)+(n-1)+n$$ $$2S=(1+n)+(2+(n-2))+\cdots+(n+1)$$ $$S=\frac{n(1+n)}{2}$$ ...
3answers
1k views

### The logic behind the rule of three on this calculation

First, to understand my question, checkout this one: Calculating percentages for taxes. Second, consider that I'm a layman in math. So, after trying to understand the logic used to get the final ...
3answers
93 views

### How to expand $(a_0+a_1x+a_2x^2+…a_nx^n)^2$?

I know you can easily expand $(x+y)^n$ using the binomial expansion. However, is there a simple summation formula for the following expansion? $$(a_0+a_1x+a_2x^2+...+a_nx^n)^2$$ I found something ...
2answers
73 views

### Show that equation has no solution in $(0,2\pi)$

Hi I want to show that the equation $2=2 \cos(x)+x \sin(x)$ has no solution in $(0,2 \pi)$. Since it is algebraically impossible to solve this equation for $x$ I wanted to ask you whether one of you ...
4answers
391 views

### Problems with Inequalities

It seems like I am facing some confusion while handling with inequalities,I was doing some tasks where it is asked to find the interval of the variable,after some steps I deduced the the necessary ...
4answers
632 views

### Proving that $\sum\limits_{i=1}^k i! \ne n^2$ for any $n$ [duplicate]

Possible Duplicate: How to prove that the number 1!+2!+3!+…+n! is never square? Show that $\displaystyle\sum\limits_{i=1}^k i!$ is never a perfect square for $k\ge4$ I could prove ...
5answers
366 views

### How to prove that $\sum\limits_{i=0}^p (-1)^{p-i} {p \choose i} i^j$ is $0$ for $j < p$ and $p!$ for $j = p$

Let $p \in \mathbf{N}$. I don't know how to prove that $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^j=0 \textrm{ for } j \in \{0,\ldots,p-1\},$$ and $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^p=p!$$ ...
2answers
891 views

### What is the best way to solve an equation involving multiple absolute values?

An absolute value expression such as $|ax-b|$ can be rewritten in two cases as $|ax-b|=\begin{cases} ax-b & \text{ if } x\ge \frac{b}{a} \\ b-ax & \text{ if } x< \frac{b}{a} \end{cases}$, ...
8answers
443 views

### How to factor quadratic $ax^2+bx+c$?

How do I shorten this? How do I have to think? $$x^2 + x - 2$$ The answer is $$(x+2)(x-1)$$ I don't know how to get to the answer systematically. Could someone explain? Does anyone have a link to ...
2answers
372 views

### Proof of an inequality: $\sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}}$ [duplicate]

Possible Duplicate: Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction How do I prove the following? \sqrt{n} < \dfrac{1}{\sqrt{1}} + \dfrac{1}{\sqrt{2}} + ...
2answers
157 views

### Maximum of the difference

What is the maximum value of $f(… f(f(f(x_{1} – x_{2}) – x_{3})-x_{4}) … – x_{2012})$ where $x_{1}, x_{2}, … , x_{2012}$ are distinct integers in the set ${1, 2, 3, …, 2012}$ and $f$ is the absolute ...
1answer
220 views

### $f(x)=x^3+ax^2+bx+c$ has roots $a,b$ and $c$

How many ordered triples of rational numbers $(a,b,c)$ are there such that the cubic polynomial $f(x)=x^3+ax^2+bx+c$ has roots $a,b$ and $c$? The polynomial is allowed to have repeated roots.
3answers
264 views

### There isn't a product operation that is commmutative on $\mathbb{R}^{n}$ that satisfies all the field axioms for $n \geq 3$.

This proof is broken down into simple easy algebra and vector questions. I would like to discuss different answers and approaches. Please see pg 162-163 on books.google.ca/books?isbn=0387290524 ...
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