Question and Answers Forum

All Questions      Topic List

Number Theory Questions

Previous in All Question      Next in All Question      

Previous in Number Theory      Next in Number Theory      

Question Number 55631 by Tawa1 last updated on 28/Feb/19

Answered by tanmay.chaudhury50@gmail.com last updated on 01/Mar/19

$${(1-x)}^{\frac{-1}{2}}$$$$=1+\frac{1}{2}x+\frac{1\times 3}{2\times 4}{x}^2+\frac{1\times 3\times 5}{2\times 4\times 6}{x}^3+...$$$$\mathit{L}HS$$$$1+\frac{4}{10}+\frac{4\times 12}{10\times 20}+\frac{4\times 12\times 20}{10\times 20\times 30}+...$$$$=1+\frac{1}{2}\left(\frac{8}{10}\right)+\frac{1\times 3}{2\times 4}\times \left(\frac{4\times 4}{5\times 5}\right)+\frac{1\times 3\times 5}{2\times 4\times 6}\left(\frac{4\times 4\times 4}{5\times 5\times 5}\right)+..$$$$=1+\frac{1}{2}\left(\frac{4}{5}\right)+\frac{1\times 3}{2\times 4}\times {\left(\frac{4}{5}\right)}^2+\frac{1\times 3\times 5}{2\times 4\times 6}{\left(\frac{4}{5}\right)}^3+..$$$$={(1-\frac{4}{5})}^{\frac{-1}{2}}$$$$={\left(\frac{1}{5}\right)}^{\frac{-1}{2}}={\left(5^{-1}\right)}^{\frac{-1}{2}}={\left(5\right)}^{\frac{1}{2}}=\sqrt{5}$$$$\mathit{R}HS$$$$2(1+\frac{1}{10}+\frac{1.3}{10.20}+\frac{1.3.5}{10.20.30}+...)$$$$=2(1+\frac{1}{2}\times \frac{1}{5}+\frac{1\times 3}{2\times 4}\times \frac{1}{5\times 5}+\frac{1\times 3\times 5}{2\times 4\times 6}\times \frac{1}{5\times 5\times 5}+..)$$$$=2[1+\frac{1}{2}\left(\frac{1}{5}\right)+\frac{1\times 3}{2\times 4}{\left(\frac{1}{5}\right)}^2+\frac{1\times 3\times 5}{2\times 4\times 6}{\left(\frac{1}{5}\right)}^3+...]$$$$=2\left[{(1-\frac{1}{5})}^{\frac{-1}{2}}\right]$$$$=2{\left[\left(\frac{4}{5}\right)\right]}^{\frac{-1}{2}}$$$$=2\times {\left(4\right)}^{\frac{-1}{2}}\times {\left(\frac{1}{5}\right)}^{\frac{-1}{2}}$$$$=2\times 2^{-1}\times {\left(5^{-1}\right)}^{\frac{-1}{2}}$$$$=2\times \frac{1}{2}\times {\left(5\right)}^{\frac{1}{2}}$$$$={\left(5\right)}^{\frac{1}{2}}=\sqrt{5}$$$$\mathit{h}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{d}...$$$$$$$$$$$$$$

Commented by Tawa1 last updated on 03/Mar/19

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

Terms of Service

Privacy Policy

Contact: info@tinkutara.com