Find the sum of $n$ terms of a series $1^{3} + 3^{3} + 5^{3} + \dots$

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Solution

$\begin{matrix} S_{n} = 1^{3} + 3^{3} + 5^{3} + . . . = n \sum r = 1 {(2 r - 1)}^{3} = n \sum r = 1 8 r^{3} - 1 - 12 r^{2} + 6 r = \frac{8 n^{2} {(n + 1)}^{2}}{4} - n - \frac{12 n (n + 1) (2 n + 1)}{6} + \frac{6 n (n + 1)}{2} \end{matrix}$

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