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Question

If a,b,c be the respective sums of the n terms, next n terms and of G.P. then prove that a,b,c are in G.P.

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Solution

Let x be the first term & r be the common ratio of the G.P
a=x(1rn)1r ………(1)
b=x(1r2n)1ra
a+b=x(1r2n)1r …..(2)
c=x(1r3n)1r(a+b)
a+b+c=x(1r3n)1r ……(3)
(2)÷(1)
a+ba=1r2n1rn
1+ba=1+rn
b=arn
(3)÷(2)
a+b+ca+b=1r3n1r2n
1+ca+b=1+rn+r2n1+rn
ca+b=r2n1+rn
ca+arn=r2n1+rn
ca(1+rn)=r2n1+rn
c=ar2n
We can say that a,arn,ar2n are in G.P.
a, b, c are in G.P.

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