Question

Find the sum to n terms of the series 7 + 77 + 777 + ............

Answer 1

$7,77,777,7777.............$ to n terms.

Sn=$7+77+777+7777$+...........to n terms

introducing $9$

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

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

=$\frac{7}{9}$ [$10+100+1000$ + ... n terms - ($1+1+1+1$+......... n terms)

= $\frac{7}{9}$ $[10+100+1000$+.......................n terms -n ]

Here $a=10,r=10$

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

= $\frac{70}{81}$$({10}^n-1)-n$