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Question Number 154896 by maged last updated on 22/Sep/21

$$findthefollowing?please$$$$S=\underset{k=1}{\sum ^n}\frac{1}{k(k+1)(k+2)}$$

Commented by benhamimed last updated on 23/Sep/21

$$\mathrm{\Sigma }\frac{1}{k(k+1)(k+2)}=\mathrm{\Sigma }\frac{1}{2}(\frac{1}{k}-\frac{2}{k+1}+\frac{1}{k+2})$$$$=\frac{1}{2}(1-\frac{2}{2}+\frac{1}{3}+\frac{1}{2}-\frac{2}{3}+\frac{1}{4}+\frac{1}{3}-\frac{2}{4}+\frac{1}{5}+...$$$$\frac{1}{n-1}-\frac{2}{n}+\frac{1}{n+1}+\frac{1}{n}-\frac{2}{n+1}+\frac{1}{n+2})$$$$=\frac{1}{2}(\frac{1}{2}-\frac{1}{n+1}+\frac{1}{n+2})$$

Commented by maged last updated on 23/Sep/21

$$Thankyou$$

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