A geometric sequence begins:
The sum of the first seven terms is: .
Again, because there are only seven terms, it isn’t unreasonable to go ahead and add them together. But let’s do it in a way that can be generalised into a formula.
First, lets give the sum a name. As we are adding seven terms, we call the sum .
Next, we multiply both sides by the common ratio which is .
Then we subtract from :
Notice that on the left hand side, we just have , and on the right hand side all but two of the terms cancel out.
This is a tidy calculation to preform: we find that .
To generalize, let’s write the first seven terms of a geometric sequence:
Notice that the first term is not multiplied by and that the last term is the term.
Now, if we multiply both sides by the common ratio we have:
Subtracting like we did in the example above we have
Again, all but two terms on the right hand side cancel out.
Let’s divide through by :
Now take out the common factor on the numerator:
Finally, we are using 7 terms but we could be adding any number of terms, say terms. Let’s replace the 7 with :
And there is the formula.
Notice that to use this formula, it is not necessary to write out the terms of the sequence. The information required is , the number of terms; the first term and the common ratio.
Find eleven questions to practice this math at the end of this mathisfun page.
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