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

sum of first 8 terms
= a(r^8 - 1)/(r-1)
sum of first 4 terms
= a(r^4 - 1)/(r-1

( a(r^8 - 1)/(r-1) ) / ( a(r^4 - 1)/(r-1) ) = 97/81
(r^8 - 1) / (r^4 - 1) = 97 / 81
81r^8 - 81 = 97r^4 - 97
81^8 - 97r^4 + 16 = 0

let r^4 = x
81x^2 - 97x+16 = 0
x = (97 ± √4225)/162
= (97 ± 65)/162
= 1 or 16/81

r^4 = 1 , r = ± 1
r^4 = 16/81 , r = ± 2/3