Evaluate: _ k = 1 ^ 11 ( 2 + 3 ^ k ) - Vidyadip

Question

Evaluate:$\sum \limits_ { k = 1 } ^ { 11 } \left( 2 + 3 ^ { k } \right)$

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Answer

Given:$\sum _ { k = 1 } ^ { 11 } \left( 2 + 3 ^ { k } \right)$
$=\left(2+3^1\right)+\left(2+3^2\right)+\left(2+3^3\right)+\left(2+3^{11}\right)$
$=(2+2+2+\ldots \ldots . .11 \text { times })+\left(3+3^2+3^3+\ldots \ldots . .+3^{11}\right)$
$=22+\left(3+3^2+3^3+\ldots \ldots . .+3^{11}\right) \ldots \ldots \ldots . \text { (i) }$
Here $3,3^2, 3^3 \ldots \ldots . ., 3^{11}$ is in G.P.
$\therefore$a = 3 and r = $\frac { 3 ^ { 2 } } { 3 } = 3$
$\mathrm { S } _ { n } = \frac { 3 \left( 3 ^ { 11 } - 1 \right) } { 3 - 1 } = \frac { 3 } { 2 } \left( 3 ^ { 11 } - 1 \right)$
Putting the value of $S_n$ in eq. (i), we get $\sum _ { k = 1 } ^ { 11 } \left( 2 + 3 ^ { k } \right) = 22 + \frac { 3 } { 2 } \left( 3 ^ { 11 } - 1 \right)$

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