Question

In a $G.P$. of even number of terms, the sum of all terms is $5$ times the sum of the odd terms. The common ratio of the $G.P$. is

• A. $\frac{-4}{5}$
• B. $\frac{1}{5}$
• C. $4$
• D. $-5$

Answer 1

Answer: (C)

$4$

Answer 2

Let us consider a $G.P. a, ar, ar^{2},...$ with $2 n$ terms, we have $\frac{a\left(r^{2n } -1\right)}{r-1}$ $= \frac{5a \left(\left(r^{2}\right)^{n}-1\right)}{r^{2}-1}$ (since common ratio of odd terms will be $r^2$ and number of terms will be $n$) $\Rightarrow \frac{a\left(r^{2n} -1\right)}{r-1}$ $= 5 \frac{a\left(r^{2n} -1\right)}{\left(r^{2}-1\right)}$ $\Rightarrow a\left(r+1\right) = 5a$ $\Rightarrow r= 4$