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+....+arn−1          [∵tn=arn−1]    .....(i) Also sum of term from n+1th to (2n) th term is =an+1+an+2+...+a2n =arn+arn−1+....+ar2n−1       .......(ii) Ratio of (i) and (ii) is =a+ar+ar2+....arn−1arn+arn−1+....+ar2n−1 =a(1−rn)1−rarn(1−rn)1−r      ([\because S_n = \frac{a(1 - r^n)}{1 - r}]) =1rn.

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