Mathematics - Sequence & Series Question with Solution | TestHub

For the given GP, $\mathrm{n}=64, {\mathrm{S}}_{\mathrm{n}}=7{\mathrm{S}}_{\mathrm{o}} ; \left\{\text{where} {\mathrm{S}}_{\mathrm{o}}\to \text{Sum of odd terms}\right\}$

$\Rightarrow \mathrm{a}+\mathrm{ar}+{\mathrm{ar}}^2+{\mathrm{ar}}^3+...+{\mathrm{ar}}^{63}=7\left(\mathrm{a}+{\mathrm{ar}}^2+{\mathrm{ar}}^4\dots ..+{\mathrm{ar}}^{62}\right)$

$\Rightarrow \frac{\mathrm{a}\left(1-{\mathrm{r}}^{64}\right)}{1-\mathrm{r}}=\frac{7\mathrm{a}\left(1-{\mathrm{r}}^{64}\right)}{1-{\mathrm{r}}^2}$

$\Rightarrow \left(1-{\mathrm{r}}^{64}\right)=\frac{7\left(1-{\mathrm{r}}^{64}\right)}{1+\mathrm{r}}$

$\Rightarrow 1=\frac{7}{1+\mathrm{r}}$

$\Rightarrow 1+\mathrm{r}=7$

$\Rightarrow \mathrm{r}=6$