Question

A G.P. has even number of terms . If the sum of all the terms is $5$ times the sum of the terms occupying the odd places, then the common ratio of the G.P. is

• A. $2$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $4$

Answer 1

The correct option is D $4$

Let the first term and common ratio of the G.P. be $a$ and $r$ respectively and the number of terms be $2n$, so
$S_{2n}=5(T_1+T_3+T_5+\dots +T_{2n-1})\Rightarrow a\left(\frac{r^{2n}-1}{r-1}\right)=5a\left(\frac{(r^2{)}^n-1}{r^2-1}\right)\Rightarrow r+1=5\therefore r=4$