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.

Answer 1

The sum of all terms in a $G.P$ is:

$S_n=\frac{a(r^n-1)}{r-1}\to \left(1\right)$

$\to$ Here, the $G.P$ consist of even numbers of terms So, we can takes number of terms $=2a$

$\to$ Let the $G.P$ be $a,ar,a{r}^2,a{r}^3,a{r}^4......$ to $2n$ terms

$\to$ Now finding the sum of terms occupying odd places

$\to$ The $G.P$ having even terms and the series occupying odd terms is

$a,a{r}^2,a{r}^4,....$ to $n$ terms

Here $1^{st}$ term $=a$

Common ratio $=\frac{a{r}^2}{a}=r^2$

$\to$ Sum of series occupying $S_1=\frac{a\left[\right(l^2{)}^n-1}{r^2-1}$

odd places

$S_1=\frac{a[r^{2n}-1]}{r^2-1}\to \left(2\right)$

$\to$ It is given that sum of all terms is $5$ times the sum of terms occupying odd places

$\therefore S_{2n}={55}_6$, where $S_{2n}\frac{a(r^{2n}-1}{r-1}\to \left(3\right)$