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Q. 72

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.

$3.454545...$

The given repeating decimal as a geometric series is $\sum _{k=0}^{\infty}0.45{\left(0.01\right)}^{k},$ and as the quotient of two integers reduced to lowest terms is $\frac{342}{99}.$

The given repeating decimal is $3.454545...$

The repeating decimal as a geometric series can be expressed as

$3.454545...\phantom{\rule{0ex}{0ex}}=3+0.454545...\phantom{\rule{0ex}{0ex}}=3+0.45\left(1+0.01+{\left(0.01\right)}^{2}+{\left(0.01\right)}^{3}+...\right)\phantom{\rule{0ex}{0ex}}=3+\sum _{k=0}^{\infty}0.45{\left(0.01\right)}^{k}$

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

$3.454545...\phantom{\rule{0ex}{0ex}}=3+0.454545...\phantom{\rule{0ex}{0ex}}=3+0.45\left(1+0.01+{\left(0.01\right)}^{2}+{\left(0.01\right)}^{3}+...\right)\phantom{\rule{0ex}{0ex}}=3+\sum _{k=0}^{\infty}0.45{\left(0.01\right)}^{k}\phantom{\rule{0ex}{0ex}}\mathrm{Use}{S}_{\infty}=\frac{a}{1-r}\phantom{\rule{0ex}{0ex}}=3+\frac{0.45}{1-0.01}\phantom{\rule{0ex}{0ex}}=3+\frac{0.45}{0.99}\phantom{\rule{0ex}{0ex}}=3+\frac{45}{99}\phantom{\rule{0ex}{0ex}}=\frac{342}{99}$

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