91Ó°ÊÓ

In this question convert the repeating decimal $0.818181$ into fraction with the help of geometric series.

Repeating Decimal to Fraction:

A fraction is a number that represents a portion of a whole in mathematics. It is, in other words, a two-number ratio. In contrast, a decimal is a number with a decimal point between the total number and the fractional sections. There are two kinds of decimal numbers: terminating and non-terminating decimals, as well as repeating and non-repeating decimals. When dealing with many mathematical problems, converting decimal to fractional value is favoured because fractional values may be readily simplified.

Write the repeating decimal as:

$\begin{array}{rcl}0.818181...& =& 0.81+0.0081+0.000081+...\\ & =& \frac{81}{100}+\frac{81}{10000}+\frac{81}{1000000}+...\\ & =& \underset{i=1}{\overset{∞}{∑}}\frac{81}{{10}^{2i}}\\ & =& \underset{i=1}{\overset{∞}{∑}}\frac{81}{{100}^i}\\ & & \end{array}$

Implies that,

$\begin{array}{rcl}0.818181...& =& 81\underset{i=1}{\overset{∞}{∑}}\frac{1}{{100}^i}\\ & =& 81\underset{i=1}{\overset{∞}{∑}}{\left(\frac{1}{100}\right)}^i\\ & =& 81\left[-1+\underset{i=0}{\overset{∞}{∑}}{\left(\frac{1}{100}\right)}^i\right]\\ & & \end{array}$

Solve it further,

$\begin{array}{rcl}0.818181...& =& 81\left[-1+\frac{1}{1-\left(\frac{1}{100}\right)}\right]\\ & =& 81\left(-1+\frac{100}{99}\right)\\ & =& 81\left(\frac{1}{99}\right)\\ & =& \frac{9}{11}\\ & & \end{array}$

Hence, the required answer is $0.818181....=\frac{9}{11}$.