If 2 to the power 10 + 2 to the power 9.31 + .........

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:

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

  2. $3^{11}-2^{12}$

  3. $3^{11}$

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


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)

$\therefore \mathrm{S}=3^{11}$

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