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How do you derive the geometric series formula?

How do you derive the geometric series formula?-example-1

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Let
S_i denote the
ith partial sum of a geometric sequence with common ratio
r. The sum of the first
n terms of the sequence is


S_n=a+ar+ar^2+ar^3+\cdots+ar^(n-1)+ar^n.

Suppose we multiply both sides by
r, then


rS_n=ar+ar^2+ar^3+ar^4+\cdots+ar^n+ar^(n+1)

Taking the difference makes most of the terms vanish:


S_n-rS_n=a+(ar-ar)+(ar^2-ar^2)+\cdots+(ar^n-ar^n)-ar^(n+1)

1S_n-rS_n=a-ar^(n+1)

Then solving for
S_n gives the formula.


(1-r)S_n=a\left(1-r^(n+1)\right)

S_n=a(1-r^(n+1))/(1-r)
User Jakob Svenningsson
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