Let $a_n$  be the $n^{\text{th}}$  term of G.P. If ${\displaystyle \sum _{n=1}^{100}{a}_{2n}=\alpha }$  and ${\displaystyle \sum _{n=1}^{100}{a}_{2n-1}=\beta }$ , then the common ratio of the G.P is

$\Rightarrow ar+a{r}^3+a{r}^5+\cdots +a{r}^{199}=\alpha$

$\Rightarrow ar\left[1+r^2+r^4+\cdots +r^{198}\right]=\alpha \cdots \cdots \cdots \left(1\right)$

$a_1+a_3+a_5+\cdots +a_{199}=\beta$

$\Rightarrow a+a{r}^2+a{r}^4+\cdots +a{r}^{198}=\beta$

$\Rightarrow a\left[1+r^2+r^4+\cdots +r^{198}\right]=\beta \cdots \cdots \cdots \left(2\right)$

$\frac{\left(1\right)}{\left(2\right)}\Rightarrow r=\frac{\alpha }{\beta }$