Question

Find the sum of the following series up to $n$ terms:
$5+55+555+...$
$.6+.66+.666+...$

Answer 1

$S{n}^{\prime }=5+55+555+....nterms$

$=5\left(1+11+111+.......nterms\right)$

$=\frac{5}{9}\left[9+99+999+.....nterms\right]$

$=\frac{5}{9}\left[\left(10-1\right)+\left(100-1\right)+\left(1000-1\right)+.....nterms\right]$

$=\frac{5}{9}\left[10+{10}^2+{10}^3+....nterms-n\right]$

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

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

$S{n}^{\prime }=0.6+0.66+0.666+.....nterms$

$=\frac{6}{10}+\frac{66}{100}+\frac{666}{1000}+........nterms$

$=\frac{6}{10}\left[1+\frac{11}{10}+\frac{111}{100}+....nterms\right]$

$=\frac{6}{90}\left[9+\frac{99}{10}+\frac{999}{100}+\frac{999}{100}+....nterms\right]$