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

1 Answer
Jun 7, 2016

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.