Question

The sum of the terms of an infinite G.P. is 5 and the sum of the squares of those terms is 15. Find the G.P.

Answer 1

Let the required G.P. be a, ar, ar², ar³, …
Sum to infinity of this G.P. = 5

∴ 5 = `"a"/(1 - "r")`

∴ a = 5(1 – r)                     ...(i)
Also, the sum of the squares of the terms is 15.
∴ (a² + a²r² + a²r⁴ + …) = 15

∴ 15 = `"a"^2/(1 - "r"^2)`

∴ 15(1 – r²)  = a²
∴ 15(1 – r)(1 + r) = 25 (1 – r)²       ...[From (i)]
∴  3 (1 + r) = 5(1 – r)
∴  3 + 3r = 5 – 5r
∴ 8r = 2

∴ r = `1/4`

∴ a = `5(1 - 1/4) = 5(3/4) = 15/4`

∴ Required G.P. is a, ar, ar², ar³, …

i.e., `15/4, 15/16, 15/64`, ...