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 the common ratio is

• A. 5
• B. 1
• C. 4
• D. 3

Answer 1

Answer: (C)

Let the G.P. be $a,ar,a{r}^2,..........$
$S=a+ar+a{r}^2+..........+$ to $2n$ term
$=\frac{a\left(r^{2n}-1\right)}{r-1}$
The series formed by taking term occupying odd places is $S_1=a+a{r}^2+a{r}^4+........$ to $n$ terms
$S_1=\frac{a\left[{\left(r^2\right)}^n-1\right]}{r^2-1}\Rightarrow S_1=\frac{a\left(r^{2n}-1\right)}{r^2-1}$
Now, $S=5S_1$
or $\frac{a\left(r^{2n}-1\right)}{r-1}=5\frac{a\left(r^{2n}-1\right)}{r^2-1}$
$\Rightarrow 1=\frac{5}{r+1}$
$\Rightarrow r+1=5\therefore r=4$