Let the required G.P. be a, ar, ar2 , ar3 , .....
Sum to infinity of this G.P. = 5
\(\therefore\) 5 = \(\frac{a}{1-r}\)
\(\therefore\) a = 5(1 - r).....(i)
Also, the sum of the squares of the terms is 15.
\(\therefore\) (a2 + a2r2 + a2r4 + ...) = 15
\(\therefore\) 15 = \(\frac{a^2}{1-r^2}\)
\(\therefore\) 15 (1 - r2) = a2
\(\therefore\) 15(1 - r)(1 + r) = 25(1 - r)2....[From(i)]
\(\therefore\) 3(1 + r) = 5(1 - r)
\(\therefore\) 3 + 3r = 5 - 5r
\(\therefore\) 8r = 2
\(\therefore\) r = 1/4
\(\therefore\) a = 5(1- 1/4) = 5(3/4) = 15/4
\(\therefore\) Required G.P. is a, ar, ar2, ar3,....
i.e. 15/4, 15/16, 15/64,.....
