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Q. 77

Expert-verifiedFound in: Page 616

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Express each of the repeating decimals in Exercises *71–78* as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.199999...$

The given repeating decimal as a geometric series is $\frac{1+\sum _{k=0}^{\infty}0.9{\left(0.1\right)}^{k}}{10},$ and as the quotient of two integers reduced to lowest terms is $\frac{1}{5}.$

The given repeating decimal is $0.199999...$

The given repeating decimal starts repeating after the tenths place so, to express it as a geometric series, let $y=0.199999...$

Now, multiply both the sides by *10*

$\left(10\right)y=\left(10\right)0.199999...\phantom{\rule{0ex}{0ex}}10y=1.99999...\phantom{\rule{0ex}{0ex}}y=\frac{1.99999...}{10}\phantom{\rule{0ex}{0ex}}y=\frac{1+0.99999...}{10}\phantom{\rule{0ex}{0ex}}y=\frac{1+0.9\left(1+0.1+{\left(0.1\right)}^{2}+{\left(0.1\right)}^{3}+...\right)}{10}\phantom{\rule{0ex}{0ex}}y=\frac{1+\sum _{k=0}^{\infty}0.9{\left(0.1\right)}^{k}}{10}..........\left(i\right)$

The given repeating decimal as the quotient of two integers reduced to the lowest terms can be expressed as

$0.199999...\phantom{\rule{0ex}{0ex}}=\frac{1+\sum _{k=0}^{\infty}0.9{\left(0.1\right)}^{k}}{10}\mathrm{Use}\left(i\right)\phantom{\rule{0ex}{0ex}}\mathrm{Now},\mathrm{use}{S}_{\infty}=\frac{a}{1-r}\phantom{\rule{0ex}{0ex}}=\frac{1+\frac{0.9}{1-0.1}}{10}\phantom{\rule{0ex}{0ex}}=\frac{1+\frac{0.9}{0.9}}{10}\phantom{\rule{0ex}{0ex}}=\frac{2}{10}\phantom{\rule{0ex}{0ex}}=\frac{1}{5}$

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