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Summation by Sigma Method

Evaluate 1....

Question

Evaluate $1.6 + 2.9 + 3.12 + . . . + n (3 n + 3) =$

n (n + 1) (n + 2)

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(n + 1) (n + 2) (n + 3)

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(n + 2) (n + 3) (n + 4)

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(n - 1) n (n + 1)

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Solution

The correct option is B $n (n + 1) (n + 2)$
$\sum_{k = 1}^{n} k (3 k + 3) = 3 \sum_{k = 1}^{n} k^{2} + 3 \sum_{k = 1}^{n} k$
$= 3 \times \frac{n (n + 1) (2 n + 1)}{6} + 3 \times \frac{n (n + 1)}{2}$
$= \frac{n (n + 1)}{2} (2 n + 1 + 3)$
$= n (n + 1) (n + 2)$

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