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Question

If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively 2SS1S2+S1 and S2S1S2+S1.


Solution

S=a+ar+ar2+ar3+....

S=a1r        .......(1)

S1=a2+a2r2+a2r4+a2r6.....

S1=a21r2        ........(2)

S2=a2(1r)2

S2=S1(1r2)1r2

(1r)S2=S1(1+r)

S2S2r=S1+S1r

S1r+S2r=S2S1

r=S2S1S1+S2

Put r in equation (1)

S(1r)=a

a=S[1S2S1S2+S1]

a=S[S2+S1S2+S1S2+S1]

a=2SS1S2+S1

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