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Question Number 166707 by amin96 last updated on 25/Feb/22

$$\frac{8}{1\times 5\times 9}+\frac{8}{5\times 9\times 13}+\frac{8}{9\times 13\times 17}+\dots +\frac{1}{41\times 45\times 49}=?$$$$$$$$$$$$\mathit{b}\mathit{y}\mathit{M}.\mathit{A}$$

Answered by mahdipoor last updated on 25/Feb/22

$$\frac{8}{n\times (n+4)\times (n+8)}=\frac{1/4}{n}-\frac{1/2}{n+4}+\frac{1/4}{n+8}$$$$\Rightarrow \frac{8}{1\times 5\times 9}+\frac{8}{5\times 9\times 13}+\frac{8}{9\times 13\times 17}+...$$$$+\frac{8}{41\times 45\times 49}=$$$$(\frac{0.25}{1}-\frac{0.5}{5}+\frac{0.25}{9})+(\frac{0.25}{5}-\frac{0.5}{9}+\frac{0.25}{13})+$$$$+(\frac{0.25}{9}-\frac{0.5}{13}+\frac{0.25}{17})+...+(\frac{0.25}{41}-\frac{0.5}{45}+\frac{0.25}{49})=$$$$=(\frac{0.25}{1}-\frac{0.5}{5})+\left(\frac{0.25}{5}\right)+\left(\frac{0.25}{45}\right)+(-\frac{0.5}{45}+\frac{0.25}{49})=$$$$0.2-\frac{1}{45\times 49}$$$$$$

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