Question

Let $a_n$ be the $n^{th}$ term of a $G.P.$ of positive numbers. Let $\sum _{n=1}^{100}{a}_{2n}=\alpha$ & $\sum _{n=1}^{100}{a}_{2n-1}=\beta$ such that $\alpha \ne \beta$ Then the common ration of the $G.P.$ is

• A. $\frac{\alpha }{\beta }$
• B. $\frac{\beta }{\alpha }$
• C. $\sqrt{\frac{\alpha }{\beta }}$
• D. $\sqrt{\frac{\beta }{\alpha }}$

Answer 1

Answer: (A)

$\frac{\alpha }{\beta }$

Let $R$ be the common ratio of the G.P

Then $a_{2n}=a_{2n-1}R$

$\therefore \sum _{r=1}^{100}{a}_{2n}=\sum _{r=1}^{100}{a}_{2n-1}R=R\sum _{r=1}^{100}{a}_{2n-1}$
$\Rightarrow \alpha =R\beta$

$\Rightarrow R=\frac{\alpha }{\beta }$