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Question Number 46461 by Tawa1 last updated on 26/Oct/18

$$\mathrm{Find}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{nth}\mathrm{term}\mathrm{of}\mathrm{the}\mathrm{series}:\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+...$$

Answered by tanmay.chaudhury50@gmail.com last updated on 26/Oct/18

$$S_n=1-\frac{1}{2}+1-\frac{1}{2^2}+1-\frac{1}{2^3}+1-\frac{1}{2^4}+...$$$$S_n=(1+1+1...)-(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...)$$$$S_n=n-\left\{\frac{1}{2}\left(\frac{1-\frac{1}{2^n}}{1-\frac{1}{2}}\right)\right\}$$$$S_n=n-1+\frac{1}{2^n}$$$$$$

Commented by Tawa1 last updated on 26/Oct/18

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

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