Question

Sum upto $n$ the series
$6+66+666+.....$

Answer 1

$S=6+66+666+....$
$=\frac{6}{9}[9+99+999+.....]$
$=\frac{6}{9}\left[\right(10-1)+({10}^2-1)+({10}^3-1)+....]$
$=\frac{6}{9}\left[\right(10+{10}^2+{10}^3+.......-\left)(-1+1+1+....)\right]$
$=\frac{6}{9}\left[10.\frac{{10}^{n-1}}{10-1}n\right]=\frac{6}{81}[{10}^{n+1}-9n-10]$