Question

Evaluate $7+77+777+...$ upto $n$ terms.

Answer 1

Let $S=7+77+777+...$ upto $n$ terms.
$=\frac{7}{9}[9+99+999+...]$
$=\frac{7}{9}[(10-1)+({10}^2-1)({10}^3-1)+...+uptonterm]$
$=\frac{7}{9}[10+{10}^2+{10}^3+...+{10}^n-n]$
$=\frac{7}{9}(\frac{10({10}^n-1)}{9}-n)$
$=\frac{7}{81}[{10}^{n+1}-9n-10]$