Find the value of S for an infinite geometric series with r=1/8 and a=10

Question

The answer is -15

Answer 1

S for the GP is 10+10/8+10/8^2+...+10/8^n. (1/8)S=10/8+10/8^2+...+10/8^(n+1).

S-(1/8)S=10, because all terms cancel out except for 10 and 10/8^(n+1), and as n becomes very large this last term becomes almost zero. So (7/8)S=10 as n approaches infinity, and S=80/7=11 3/7=11.42857 approx. It's clear that the sum cannot be negative, and it must be bigger than 10 because of the first term and the fact that all terms are positive.