Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 107187 by Dwaipayan Shikari last updated on 09/Aug/20

$$\mathrm{Prove}\mathrm{that}$$$$\sqrt{8}=1+\frac{3}{4}+\frac{3.5}{4.8}+\frac{3.5.7}{4.8.12}+......$$

Commented by john santu last updated on 09/Aug/20

$$\sqrt{8}=\sqrt{9-1}={(9-1)}^{\frac{1}{2}}$$$$\mathrm{consider}{(9-1)}^{\mathrm{n}}=\underset{\mathrm{k}=0}{\sum ^{\mathrm{n}}}{\mathrm{C}}_{\mathrm{k}}^{\mathrm{n}}{\left(9\right)}^{\mathrm{n}-\mathrm{k}}{(-1)}^{\mathrm{k}}$$$$={\left(9\right)}^{\mathrm{n}}+\mathrm{n}{\left(9\right)}^{\mathrm{n}-1}(-1)+\frac{\mathrm{n}.(\mathrm{n}-1)}{2!}{\left(9\right)}^{\mathrm{n}-2}{(-1)}^2$$$$+\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)}{3!}{\left(9\right)}^{\mathrm{n}-3}{(-1)}^3+...$$$$\mathrm{put}\mathrm{n}=\frac{1}{2}$$

Commented by Dwaipayan Shikari last updated on 09/Aug/20

$${(1+1)}^{\frac{3}{2}}=1+\frac{3}{2}+\frac{3}{8}+(-\frac{3.}{8.6})+..\frac{3}{16.24}+...$$$$=1+\frac{3}{4}+\frac{3}{4}+\frac{3}{8}-\frac{3}{8.6}+..$$$$=1+\frac{3}{4}+\frac{3.12}{4.8}-\frac{3}{8.6}+.$$$$=1+\frac{3}{4}+\frac{3.5}{4.8}+\frac{7}{4.8}-\frac{3}{8.6}+..$$$$=1+\frac{3}{4}+\frac{3.5}{4.8}+\frac{30}{4.8.6}+\frac{3}{16.24}+..$$$$=1+\frac{3}{4}+\frac{3.5}{4.8}+\frac{3.5.4}{4.8.12}+\frac{3}{16.24}+..$$$$=1+\frac{3}{4}+\frac{3.5}{4.8}+\frac{3.5.7}{4.8.12}-\frac{3.5.3}{4.8.6}+\frac{3}{16.24}+...$$$$=1+\frac{3}{4}+\frac{3.5}{4.8}+\frac{3.5.7}{4.8.12}+.....+\mathrm{C}$$$$$$

Commented by ZiYangLee last updated on 10/Aug/20

$$\mathrm{Why}\frac{3\cdot 12}{4\cdot 8}=\frac{3\cdot 5}{4\cdot 8}+\frac{7}{4\cdot 8}???$$

Answered by ZiYangLee last updated on 10/Aug/20

$$\sqrt{8}$$$$=2^1\cdot 2^{\frac{1}{2}}$$$$={\left(\frac{1}{2}\right)}^{-1}\cdot {\left(\frac{1}{2}\right)}^{-\frac{1}{2}}$$$$={(1-\frac{1}{2})}^{-1}\cdot {(1-\frac{1}{2})}^{-\frac{1}{2}}$$$$={(1-\frac{1}{2})}^{-\frac{3}{2}}$$$$=1+\frac{(-\frac{3}{2})}{1!}(-\frac{1}{2})+\frac{(-\frac{3}{2})(-\frac{5}{2})}{2!}{(-\frac{1}{2})}^2+\frac{(-\frac{3}{2})(-\frac{5}{2})(-\frac{7}{2})}{3!}{(-\frac{1}{2})}^3+......$$$$=1+\frac{3}{4}+\frac{3\cdot 5}{4\cdot 8}+\frac{3\cdot 5\cdot 7}{4\cdot 8\cdot 12}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com