Question

Find the sum of first n terms of the series.
$0.5+0.55+0.555+....$.

Answer 1

Given,

$S_n=0.5+0.55+0.555+.....$

$=5(0.1+0.11+0.111+.....)$

$=\frac{5}{9}(0.9+0.99+0.999+.....)$

$=\frac{5}{9}\left[\right(1-0.1)+(1-0.01)+(1-0.001)+........]$

$=\frac{5}{9}[n-\left(\frac{{10}^{-1}(1-({10}^{-1}{)}^n)}{1-{10}^{-1}}\right)]$

$\therefore S=\frac{5}{9}[n-\frac{1}{9}(1-\frac{1}{{10}^n})]$