Question

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

Short answer

\( 0.8181 \ldots = \frac{81}{99} \)

Step-by-step solution

Express the decimal as a geometric series

Let \( x = 0.8181 \ldots \). Multiplying this by 100 gives \( 100x = 81.8181 \ldots \) . Subtracting the first equation from this results in \( 99x = 81 \), so \( x = \frac{81}{99} \) as a geometric series this can be expressed as \( \frac{8}{10} + \frac{1}{100} + \frac{8}{10000} + \frac{1}{1000000} + \ldots \)

Find the sum of the geometric series

The sum to infinity of a geometric series can be calculated using the formula \( S = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio. In this case, \( a = \frac{8}{10} \) and \( r = \frac{1}{100} \). Substituting these values into the formula gives \( S = \frac{\frac{8}{10}}{1-\frac{1}{100}} = \frac{81}{99} \), which is the ratio of two integers.