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Question Number 106515 by nimnim last updated on 05/Aug/20

$$Sumupthefollowingtonthterm:$$$$\frac{5}{3}+\frac{10}{8}+\frac{17}{15}+\frac{26}{24}+........$$

Answered by Dwaipayan Shikari last updated on 05/Aug/20

$$S=1+\frac{2}{3}+1+\frac{2}{8}+1+\frac{2}{15}+1+\frac{2}{24}+..$$$$S=(1+1+1+...n)+2(\frac{1}{3}+\frac{1}{8}+\frac{1}{15}+...)$$$$S=n+2\underset{n=1}{\sum ^n}\frac{1}{n(n+2)}$$$$S=n+2\underset{n=1}{\sum ^n}\frac{n+1}{n(n+2)(n+1)}$$$$S=n+2\sum ^n\frac{1}{(n+1)(n+2)}+\frac{1}{n(n+2)(n+1)}$$$$S=n+2\sum ^n\frac{1}{n+1}-\frac{1}{n+2}+\sum ^n\frac{n+2-n}{n(n+2)(n+1)}$$$$S=n+2(\frac{1}{2}-\frac{1}{n+2})+\sum ^n\frac{1}{n(n+1)}-\sum ^n\frac{1}{(n+1)(n+2)}$$$$S=n+(1-\frac{2}{n+2})+(1-\frac{1}{n+1})-\sum ^n\frac{1}{n+1}-\frac{1}{n+2}$$$$S=n+(1-\frac{2}{n+2})+(1-\frac{1}{n+1})-\frac{1}{2}+\frac{1}{n+2}$$$$S=n+\frac{3}{2}-\frac{1}{n+2}-\frac{1}{n+1}$$$$$$

Commented by nimnim last updated on 06/Aug/20

$$ThankyouSir.$$

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