The common ratio of a geometric progression is 1/2 , the fifth term is 1/80 , and the sum of all of its terms is 127/320 . Find the number of terms in the progression.

Answer 1

term5 is 1/80 and r = 1/2
so
a(1/2)^4 = 1/80
a(1/16) = 1/80
a = 1/5
so the terms are: 1/5, 1/10, 1/20, 1/40, 1/80, ....

sum(n) = 127/320
(1/5)((1/2)^n - 1) / (1/2 - 1) = 127/320
(-2/5)((1/2)^n -1) = 127/320
(1/2)^n - 1 = -635/640 = -127/128
(1/2)^n = 1 - 127/128 = 1/128 = (1/2)^7
n = 7 , there were 7 terms