Answer
$\dfrac{5}{18}$
Work Step by Step
Here, we have $a_n= a_1 r^{n-1}$ for the Geometric series.
Re-arrange the given series as: $0.2777...=0.2 +0.07 +0.07(0.1) +0.07(0.1)^2 +....$
First term $a_1=0.07$ and Common ratio $r=0.1$
The sum of an infinite Geometric Series can be found using: $S_n=\dfrac{a_1}{1-r}$
Thus, $S_n=0.2 +\dfrac{0.07}{1-0.1}=0.2 +\dfrac{7}{90}$
Hence, $S_n=\dfrac{18+7}{90}=\dfrac{5}{18}$
