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 first note that this is a repeating decimal where the \( 5 \) repeats indefinitely. We can separate this into two parts: \( 0.2 \) and \( 0.0\overline{5} \). 1. The value of \( 0.2 \) is simply \( \frac{2}{10} = \frac{1}{5} \). 2. For \( 0.0\overline{5} \), we can let \( x = 0.0\overline{5} \). Thus, \( 10x = 0.5\overline{5} \). By subtracting these equations, we get: \[ 10x - x = 0.5\overline{5} - 0.0\overline{5} \implies 9x = 0.5 \implies x = \frac{0.5}{9} = \frac{5}{90} = \frac{1}{18}. \] Combining these two parts, we have: \[ 0.2\overline{5} = \frac{1}{5} + \frac{1}{18}. \] Finding a common denominator, which is \( 90 \): \[ \frac{18}{90} + \frac{5}{90} = \frac{23}{90}. \] Thus, \( 0.2\overline{5} \) as a common fraction is \( \frac{23}{90} \).