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Question Number 126078 by amns last updated on 17/Dec/20

$$\mathbf{Tell}\mathbf{me}$$$$\frac{1}{2}-\frac{1}{12}-\frac{1}{20}-\frac{1}{30}-\frac{1}{42}-.....-\frac{1}{210}=?$$

Commented by Dwaipayan Shikari last updated on 17/Dec/20

$$\frac{7}{30}$$

Answered by Olaf last updated on 17/Dec/20

$$\mathrm{S}=\frac{1}{2}-\underset{n=1}{\sum ^{12}}\frac{1}{n^2+5n+6}$$$$\mathrm{S}=\frac{1}{2}-\underset{n=1}{\sum ^{12}}\frac{1}{(n+2)(n+3)}$$$$\mathrm{S}=\frac{1}{2}-\underset{n=1}{\sum ^{12}}[\frac{1}{n+2}-\frac{1}{n+3}]$$$$\mathrm{S}=\frac{1}{2}-[\frac{1}{3}-\frac{1}{15}]$$$$\mathrm{S}=\frac{1}{2}-\frac{4}{15}=\frac{7}{30}$$

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