Question

Find sum
$5+55+555+....$ to $n$ terms

Answer 1

Given,

$S_n=5+55+555+.......$

$=\frac{5}{9}(9+99+999+.......)$

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

$=\frac{5}{9}\left[\right(10+{10}^2+{10}^3+...........)-(1+1+1+.....\left)\right]$

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

$=\frac{5}{81}[{10}^{n+1}-10-9n]$