Question

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

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

Answer 1

The correct option is A $\frac{\alpha }{\beta }$

Let first term and common ratio of G.P. be $a$ and $r$ respectively.
$\sum _{n=1}^{100}{a}_{2n}=\alpha$
$\Rightarrow a_2+a_4+a_6+\cdots +a_{200}=\alpha$
$\Rightarrow ar+a{r}^3+a{r}^5+\cdots +a{r}^{199}=\alpha$
$\Rightarrow ar(1+r^2+r^4+\cdots +r^{198})=\alpha \cdots \left(1\right)$

$\sum _{n=1}^{100}{a}_{2n-1}=\beta$
$\Rightarrow a_1+a_3+a_5+\cdots +a_{199}=\beta$
$\Rightarrow a+a{r}^2+\cdots +a{r}^{198}=\beta$
$\Rightarrow a(1+r^2+\cdots +r^{198})=\beta \cdots \left(2\right)$
Dividing $\left(1\right)$ and $\left(2\right),$ we get
$r=\frac{\alpha }{\beta }$