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Question

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n+1)th to(2n)th term is 1rn.


Solution

Sum of first n term of G.P.

=a+a2+a3+....+an

=a+ar+ar2+....+arn1          [tn=arn1]    .....(i)

Also sum of term from

n+1th to (2n) th term is

=an+1+an+2+...+a2n

=arn+arn1+....+ar2n1       .......(ii)

Ratio of (i) and (ii) is

=a+ar+ar2+....arn1arn+arn1+....+ar2n1

=a(1rn)1rarn(1rn)1r      ([\because S_n = \frac{a(1 - r^n)}{1 - r}])

=1rn.

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