Question

Find the sum of first $n$ terms of the series $6+66+666+.....$

Answer 1

The sum of $n$ terms of the series $6+66+666+....$ is

$S_n=6+66+666+....+nterms$

$=6(1+11+111+....+n)$

$=\frac{6}{9}(9+99+999+....+n)$

$=\frac{6}{9}\left[\right(10-1)+({10}^2-1)+({10}^3-1)+....+n]$

$=\frac{2}{3}\left[\right(10+{10}^2+{10}^3+....+nterms)-(1+1+1+....+nterms\left)\right]$

$=\frac{2}{3}[\frac{10({10}^n-1)}{10-1}-n]$

$=\frac{2}{3}\left[\frac{10}{9}\right({10}^n-1)-n]$