The ratio of the sum of the first 8 terms of a G.P. to the sum of the first 4 terms of the same G.P. is 97:81, where the common ratio of the G.P. is a real number. What is the common ratio?

Answer 1

a(1-r^8)/(1-r) / a(1-r^4)/(1-r) = 97/81
so,
(1-r^8)/(1-r^4) = 97/81
81 - 81r^8 = 97 - 97r^4
81r^8 - 97r^4 + 16 = 0
(r^4-1)(81r^4-16) = 0
r^4 = 1 or 16/81
r = ±1 or ±2/3
Naturally, r=±1 are out, but ±2/3 both work