Question

How do you derive the geometric series formula?

Answer 1

Let
`S_i` denote the
`i`th 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)`