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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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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.

Sum
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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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Chapter 2: Sequences and Series - Exercise 2.3 [Page 34]

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