Find _ r=1 ^n 1^3+2^3+3^3+ +r^3 (r+1)^2 - Vidyadip

Question

Find $\sum_{r=1}^n \frac{1^3+2^3+3^3+\ldots+r^3}{(r+1)^2}$

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Answer

$ \begin{aligned} & \sum_{r=1}^n \frac{1^3+2^3+3^3+\ldots+r^3}{(r+1)^2} \\ = & \sum_{r=1}^n \frac{r^2(r+1)^2}{4} \times \frac{1}{(r+1)^2} \end{aligned} $ $ \begin{aligned} & =\frac{1}{4} \sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^2 \\ & =\frac{1}{4} \cdot \frac{\mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{6} \\ & =\frac{\mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{24} \end{aligned} $

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