Sum the series to $n$ terms $2.4.6 + 4.6.8 + 6.8.10 + . . . .$

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Solution

$2.4.6 + 4.6.8 + . . . . .$
$a_{n} = 2 n (2 n + 2) (2 n + 4)$
$= 8 n (n + 1) (n + 2)$
$s u m o f t h e s e r i e s = n \sum n = 1 a n$
$= \sum 8 n (n + 1) (n + 2)$
$= 8 \sum n (n^{2} + 3 n + 2)$
$= 8 [\sum n^{3} + 3 \sum n^{2} + 2 \sum n]$

$= 8 ⎡ ⎣ {\frac{n (n + 1)}{2}}^{2} + 3 \frac{n (n + 1) (2 n + 1)}{6} + \frac{2 n (n + 1)}{2} ⎤ ⎦$

$= 4 n (n + 1) [\frac{n^{2} + n + 4 n + 2 + 4}{2}]$

$= 2 n (n + 1) [n^{2} + 5 n^{2} + 6]$
$= 2 n (n + 1) (n + 2) (n + 3)$

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