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Question
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
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Solution
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, ...`
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