Question

$7+77+777+...n$ terms $=-\frac{7n}{k}+\frac{70}{m^2}({10}^n-1)$
Find $\frac{k}{m}$.
(Note : $k$ and $m$ are positive integers)

Answer 1

$S=7+77+777+...+n$ terms.
$\frac{9}{7}S=9+99+999+...$
$\frac{9}{7}S=(10-1)+(100-1)+...+({10}^n-1)$
$\frac{9}{7}S=(10+100+1000+...+{10}^n)-n$
$\frac{9}{7}S=10\left(\frac{{10}^n-1}{9}\right)-n$
$S=\frac{70}{9}\left(\frac{{10}^n-1}{9}\right)-\frac{7n}{9}$
$S=\frac{70}{81}({10}^n-1)-\frac{7n}{9}$
Comparing terms,
$k=9$ and $m=9$
$\frac{k}{m}=1$