Question

If 2¹⁰ + 2⁹ . 3 + 2⁸. 3² + 2⁷ .... 3¹⁰ =S.2¹¹ then S = ?

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

${2}^{10} + {2}^{9} . \: 3 + {2}^{8} . {3}^{2} + ........ + {3}^{10}$

$= \displaystyle \sum\limits_{r=0}^{10} {2}^{10 - r} \times {3}^{r}$

$= {2}^{10} \times \displaystyle \sum\limits_{r=0}^{10} {2}^{ - r} \times {3}^{r}$

$= {2}^{10} \times \displaystyle \sum\limits_{r=0}^{10} { \bigg( \frac{3}{2} \bigg) }^{r}$

It is a Geometric Progression with first term = 1

Common Difference

$= \frac{3}{2}$

Number of terms = 11

So

${2}^{10} + {2}^{9} . \: 3 + {2}^{8} . {3}^{2} + ........ + {3}^{10}$

$= {2}^{10} \times \displaystyle \sum\limits_{r=0}^{10} { \bigg( \frac{3}{2} \bigg) }^{r}$

$= {2}^{10} \times \displaystyle 1 \times \frac{{ \bigg( \frac{3}{2} \bigg) }^{11} - 1}{ \frac{3}{2} - 1 }$

$= {2}^{10} \times \displaystyle 1 \times \frac{{ \bigg( \frac{3}{2} \bigg) }^{11} - 1}{ \frac{1}{2} }$

$= {2}^{11} \times \displaystyle \{ {\bigg( \frac{3}{2} \:\bigg)}^{11} - 1 \}$

Hence S =

$\displaystyle \: {\bigg( \frac{3}{2} \:\bigg)}^{11} - 1$

$\displaystyle \: = \frac{ {3}^{11} }{ {2}^{11} } - 1$

Answer 2

Answer:

B is the correct answer

Step-by-step explanation:

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