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:$\sum _ { k = 1 } ^ { 11 } \left( 2 + 3 ^ { k } \right)$
= (2 + 3¹) + (2 + 3²) + (2 + 3³) + (2 + 3¹¹)
= ( 2 + 2 + 2 +........11 times) + (3 + 3² + 3³ +....... +3¹¹)

= 22 + (3 + 3² + 3³ +....... +3¹¹) ……….(i)

Here 3, 3²,3³ ....... ,3¹¹is 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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