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Mathematics
OpenStudy (anonymous):

what formula do I use for What is the sum of an 8–term geometric series if the first term is –11, the last term is 180,224, and the common ratio is –4??

OpenStudy (anonymous):

-11 x -4 = ? x -4 = ? x -4 = ? x -4 = ? x -4 = ? x -4 = ? x -4 = ? x -4 = 180,224

OpenStudy (anonymous):

Thanks but that isnt in my answer choices

OpenStudy (anonymous):

but do you understand the reasoning? 8 times multiplying by -4. What are you answers?

OpenStudy (anonymous):

no, I just need a formula, thats all

OpenStudy (anonymous):

Look at this. Let's call \(S\) the sum of the first \(n\) terms with first term \(a_1\) and common ratio \(C\). \[\huge S=a_1+Ca_1+C^2a_1+\cdots+C^{n-1}a_1\]Let us now multiply both sides by \(C\).\[\huge CS=Ca_1+C^2a_1+C^3a_1 +\cdots+C^{n}a_1\]Now, we will subtract the second equation from the first. All but two terms cancel, no matter how many terms we began with.\[\huge S-CS=a_1-C^na_1=(1-C^n)a_1\]This implies the following, if we solve for \(S\).\[\huge \boxed{S=\dfrac{1-C^n}{1-C}\ a_1}\]

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