Find the sum of 2n terms of the series $5^{3} + {4.6}^{3} + 7^{3} + 7^{3} + {4.8}^{3} + 9^{3} + {4.10}^{3} + . . . .$

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Solution

$S_{2 n} = (n$ odd terms $) + (n$ even terms $)$
$= (5^{3} + 7^{3} + 9^{3} + . . . . n t e r m s) + 4 (6^{3} + 8^{3} + 10^{3} + . . . . n t e r m s)$
$= n \sum r = 1 {(2 r + 3)}^{3} + 4 Σ {(2 r + 4)}^{3}$
$= n \sum r = 1 [8 r^{3} + 36 r^{2} + 54 r + 27] + 48 [r^{3} + 2 r^{2} + 4 r + 8]$
$= n \sum r = 1 [40 r^{3} + 228 r^{2} + 438 r + 283]$
Now put $r = 1, 2, 3, . . . . . n$ add and use the formula for $Σ n, Σ n^{2}, Σ n^{3}$
$= n [10 n^{3} + 96 n^{2} + 363 n + 540]$

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