First slide

Series of natural numbers

Question

$The sum of the series \frac{3 \times 1^{3}}{1^{2}} + \frac{5 (1^{3} + 2^{3})}{1^{2} + 2^{2}} + \frac{7 (1^{3} + 2^{3} + 3^{3})}{1^{2} + 2^{2} + 3^{2}} + \dots + upto to 10^{th} term is$

Moderate

Solution

$\begin{array}{l} T_{n} = \frac{(2 n + 1) {[\frac{n (n + 1)}{2}]}^{2}}{\frac{n (n + 1) (2 n + 1)}{6}} = \frac{3}{2} (n^{2} + n) \\ S_{10} = \sum T_{10} = \frac{3}{2} (\sum_{1}^{10} n^{2} + \sum_{1}^{10} n) = \frac{3}{2} [\frac{10 \times 11 \times 21}{6} + \frac{10 \times 11}{2}] \\ = 660 \end{array}$

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