se the formula for \( S_{\infty} \) of a geometric series to write \( 0,2 \dot{5} \) as common fraction

To express \( 0.2\overline{5} \) as a common fraction, we start by recognizing that \( 0.2\overline{5} = 0.255555...\). We can decompose this number into two parts: \( 0.2 \) and \( 0.05555...\). First, \( 0.2 \) can be written as \( \frac{2}{10} = \frac{1}{5} \). Next, for the repeating part \( 0.05555... \), we can use the formula for the sum of an infinite geometric series. Here, the first term \( a \) is \( 0.05 \) (which is \( \frac{5}{100} \)) and the common ratio \( r \) is \( 0.1 \). Using the formula \( S_{\infty} = \frac{a}{1 - r} \), we get: \[ S_{\infty} = \frac{0.05}{1 - 0.1} = \frac{0.05}{0.9} = \frac{5}{90} = \frac{1}{18} \] Now, combining both parts: \[ 0.2\overline{5} = 0.2 + 0.05555... = \frac{1}{5} + \frac{1}{18} \] To add these fractions, find a common denominator, which is 90: \[ \frac{1}{5} = \frac{18}{90}, \quad \frac{1}{18} = \frac{5}{90} \] Now, add them together: \[ \frac{18}{90} + \frac{5}{90} = \frac{23}{90} \] Thus, \( 0.2\overline{5} \) as a common fraction is \( \frac{23}{90} \).