Question:
If $2^{10}+2^{9} \cdot 3^{1}+2^{8} \cdot 3^{2}+\ldots .+2 \cdot 3^{9}+3^{10}=S-2^{11}$, then $S$ is equal to:
Correct Option: , 3
Solution:
$\mathrm{a}=2^{10} ; \mathrm{r}=\frac{3}{2} ; \mathrm{n}=11$ (G.P.)
$\mathrm{S}^{\prime}=\left(2^{10}\right) \frac{\left(\left(\frac{3}{2}\right)^{11}-1\right)}{\frac{3}{2}-1}=2^{11}\left(\frac{3^{11}}{2^{11}}-1\right)$
$\mathrm{S}^{\prime}=3^{11}-2^{11}=\mathrm{S}-2^{11}$ (Given)
$\therefore \mathrm{S}=3^{11}$