Prove the following
Question:

If $2^{10}+2^{9} \cdot 3^{1}+2^{8} \cdot 3^{2}+\ldots+2 \times 3^{9}+3^{10}=\mathrm{S}-2^{11}$ then $\mathrm{S}$ is equal to:

1. (1) $3^{11}-2^{12}$

2. (2) $3^{11}$

3. (3) $\frac{3^{11}}{2}+2^{10}$

4. (4) $2 \cdot 3^{11}$

Correct Option: , 2

Solution:

Given sequence are in G.P. and common ratio $\frac{3}{2}$

$\therefore \frac{2^{10}\left(\left(\frac{3}{2}\right)^{11}-1\right)}{\left(\frac{3}{2}-1\right)}=S-2^{11}$

$\Rightarrow 2^{10} \frac{\left(\frac{3^{11}-2^{11}}{2^{11}}\right)}{\frac{1}{2}}=S-2^{11}$

$\Rightarrow 3^{11}-2^{11}=S-2^{11} \Rightarrow S=3^{11}$