Question

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

• A. $3^{11}$
  
• B. $\frac{3^{11}}{2}+2^{10}$
  
• C. $2\cdot 3^{11}$
  
• D. $3^{11}-2^{12}$
  

Answer 1

The correct option is A $3^{11}$


Let
$S^{\prime }=2^{10}+2^9{3}^1+2^8{3}^2+\cdots +2\cdot 3^9+3^{10}$
$\frac{3\times S^{\prime }}{2}=2^9\times 3^1+2^8\cdot 3^2+\cdots +3^{10}+\frac{3^{11}}{2}$
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$\Rightarrow \frac{-S^{\prime }}{2}=2^{10}-\frac{3^{11}}{2}$
$\Rightarrow S^{\prime }=3^{11}-2^{11}$
Now, $S^{\prime }=S-2^{11}$
Therefore, $S=3^{11}$