$3.6 + 4.7 + 5.8 + . . . (n - 2) t e r m s =$

n^{3} + n^{2} + n + 2

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\frac{2 n^{3} + 12 n^{2} + 10 n - 84}{6}

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2 n^{3} + 12 n^{2} + 10 n + 84

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2 n^{3} - 12 n^{2} + 10 n - 84

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Solution

The correct option is B $\frac{2 n^{3} + 12 n^{2} + 10 n - 84}{6}$
General term : $(2 + k) (5 + k)$ where $k = 1$ to $n - 2$

$n - 2 \sum k = 1 (2 + k) (5 + k)$

$= n - 2 \sum k = 1 (k^{2} + 7 k + 10)$

$= \frac{(n - 2) (n - 1) (2 n - 3)}{6} + \frac{7 (n - 2) (n - 1)}{2} + 10 (n - 2)$

$= \frac{n - 2}{6} ((n - 1) (2 n - 3) + 21 (n - 1) + 60)$

$= \frac{2 n^{3} + 12 n^{2} + 10 n - 84}{6}$

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