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Question Number 185760 by mathlove last updated on 27/Jan/23

$$A=2\times {10}^0+{10}^{-1}+6\times {10}^{-2}+6\times {10}^{-3}+6\times {10}^{-4}+....$$$$\frac{A}{13}=?$$

Answered by Rasheed.Sindhi last updated on 27/Jan/23

$$A=2+{10}^{-1}+6({10}^{-2}+{10}^{-3}+{10}^{-4}...)$$$$=21/10+6\left(\frac{1/100}{1-1/10}\right)$$$$=\frac{21}{10}+6\left(\frac{1}{100}\times \frac{10}{9}\right)$$$$=\frac{21}{10}+\frac{1}{15}=\frac{63+2}{30}=\frac{65}{30}=\frac{13}{6}$$$$\frac{A}{13}=\frac{13}{13\times 6}=\frac{1}{6}$$

Commented by JDamian last updated on 27/Jan/23

$$2+{10}^{-1}\ne \frac{21}{2}\Rightarrow \frac{947}{90}\ne 2.1\hat{6}=A$$$$2+{10}^{-1}=\frac{21}{10}$$

Commented by Rasheed.Sindhi last updated on 27/Jan/23

$$\mathcal{T}hankssir!Mymistakes.I^{\prime }vecorrectednow.$$

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