Question

How do you find the sum of the infinite geometric series 36+24+16+...?

Answer 1

`108`

Explanation:

In any geometric series, we need two pieces of information to find its sum to the `n`-th term, and thus the sum to infinity.

Firstly, we need the first term, `a`. This is obviously `36` in this expression.

Secondly, we need the common ratio, `r`,. We find this by taking `24/36 = 2/3` and confirm that this is the common ratio by taking `16/24 = 2/3`.

Thus, `a=36` and `r=2/3`. Also note that `abs(r)=2/3<1`.

We then apply the formula for sum of geometric series to the `n`-th term when `abs(r) < 1`: `S_n = (a(1-r^n))/(1-r) = (36(1-(2/3)^n))/(1-(2/3))=108(1-(2/3)^n)`

The sum to infinity, `S_infty` is defined to be the limit of `S_n` as `n-> infty`.

`n -> infty`, `(2/3)^n -> 0`, `S_n -> 108 - 0 = 108`, `S_infty = 108`.