Question

A G. P consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.

Answer 1

Let the number of terms be 2n.

Let a, ar, $a{r}^2$, ......, $a{r}^{2n-1}$, be in G.P.

Then, $S_{2n}=\frac{a\left(r^{2n-1}\right)}{r-1}$

Now, the number of odd terms = n and common ratio $=r^2$

$\therefore S_n=\frac{a\left[\right(r^2{)}^n-1]}{r^2-1}=\frac{a(r^{2n}-1)}{r^2-1}$

$\because S_{2n}=5.S_n$ [Given]

$\therefore \frac{a(r^{2n}-1)}{r-1}=\frac{5a(r^{2n}-1)}{r^2-1}$

$\Rightarrow r+1=5$

$\Rightarrow r=5-1=4$