Question

Let the sum of $n$ terms of the series, $5+55+555+.....$ be $S_n=\frac{50}{81}(k^n-1)-\frac{5}{9}n$. Find k ?

Answer 1

Let $S_n=5+55+555+.............n$ terms
$=5\{1+11+111+....nterms\}$
$=\frac{5}{9}\{9+99+999+.....nterms\}$
$=\frac{5}{9}\left\{\right(10-1)+(100-1)+(1000-1)+.....nterms\}$
$=\frac{5}{9}\left\{\right(10+100+1000+.....nterms)-(1+1+1+......nterms\left)\right\}$
$=\frac{5}{9}\left[\frac{10({10}^n-1)}{10-1}-n\right]$
$=\frac{50}{81}({10}^n-1)-\frac{5}{9}n$