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Question

If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is 92, then write its first term and common difference.

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Solution

Let us take a G.P. whose first term is a and common difference is r.
S=a1-r a1-r= 3 .......iAnd, sum of the terms of the G.P. a2, ar2, ar22, ... :S'= a21-r2 a21-r2 =92 .......ii2a2 = 9 1-r2 231-r2 = 9 - 9r2 From i181+r2-2r = 9-9r218-9 + 18r2+9r2 -36r = 027r2-36r+9 = 039r2-12r+3= 0 9r2-12r+3= 09r2 -9r-3r+3= 09rr-1-3r-1=09r-3r-1=0r=13 and r = 1.But, r = 1 is not possible. r = 13Now, putting r =13 in a1-r = 3:a = 31-13a = 3×23 = 2

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