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Sequences and Series — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsSequences and Series3 Marks
Question
Evaluate:$\sum \limits_ { k = 1 } ^ { 11 } \left( 2 + 3 ^ { k } \right)$
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Answer

Given:$\sum _ { k = 1 } ^ { 11 } \left( 2 + 3 ^ { k } \right)$
$= (2 + 3^1) + (2 + 3^2) + (2 + 3^3) + (2 + 3^{11})$
$= ( 2 + 2 + 2 +........11\ \text{times}) + (3 + 3^2 + 3^3 +....... +3^{11})= 22 + (3 + 3^2 + 3^3 +....... +3^{11}) ……….(i)$
Here $3, 3^2,3^3 ....... ,3^{11}$  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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