Question

sum to first n terms.:
$7,77,777,7777,.......$

Answer 1

$7,77,777,...$

$S_n=7+77+777+...$

$=7[1+11+111+...]$

$=\frac{7}{9}[9+99+999+...]$

$=\frac{7}{9}\left[\right(10-1)+(100-1)+(1000-1)+...]$

$=\frac{7}{9}\left[\right(10+100+1000+...aG.P.)-(1+1+...ntimes\left)\right]$

$=\frac{7}{9}\left[10\right(\frac{1-{10}^n}{1-10})-n]$

$=\frac{7}{9}\left[10\right(\frac{1-{10}^n}{-9})-n]$

$=\frac{7}{9}\left[10\right(\frac{{10}^n-1}{9})-n]$