(a) write the repeating decimal as a geometric series, and (b) write its sum as the ratio of two integers. $0.\stackrel{\u2015}{81}$

boitshupoO
2020-12-31
Answered

(a) write the repeating decimal as a geometric series, and (b) write its sum as the ratio of two integers. $0.\stackrel{\u2015}{81}$

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Nathanael Webber

Answered 2021-01-01
Author has **117** answers

Given decimal: $0.\stackrel{\u2015}{81}$

(a) For the repeating decimal 0.81, we can write

$0.\stackrel{\u2015}{81}=0.818181...$

$=0.81+0.0081+0.000081+0.00000081+...$

$=\frac{81}{{10}^{2}}+\frac{81}{{10}^{4}}+\frac{81}{{10}^{6}}+...$

$=\sum _{n=0}^{\mathrm{\infty}}\frac{81}{{10}^{2}}(\frac{1}{{10}^{2}}{)}^{n}$
(b) Given Series: $=\sum _{n=0}^{\mathrm{\infty}}\frac{81}{{10}^{2}}(\frac{1}{{10}^{2}}{)}^{n}$

Given series is a Geometric series with ratio$r=\frac{1}{{10}^{2}}$ and ${a}_{1}=\frac{81}{{10}^{2}}$

So the sum is$=\frac{{a}_{1}}{(1-r)}$

$=\frac{\frac{81}{100}}{(1-\frac{1}{100})}$

$=\frac{81}{99}$

$=\frac{9}{11}$

Result

$\frac{9}{11}$

(a) For the repeating decimal 0.81, we can write

Given series is a Geometric series with ratio

So the sum is

Result

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