Question

The sum of the series 7+77+777+........of n terms is .

• A. $\frac{7}{81}\left({10}^{n+1}-10-9n\right]$
• B. $\frac{7}{81}\left({10}^{n+1}+10-9n\right]$
• C. $\frac{7}{81}\left({10}^n-10-9n\right]$
• D. $\frac{7}{81}\left({10}^n+10-9n\right]$

Answer 1

The correct option is A $\frac{7}{81}\left({10}^{n+1}-10-9n\right]$

$7+77+777+.......+tonterms=7(1+11+111+....+tonterms)=\frac{7}{9}\left(9+99+999+......+tonterms\right)=\frac{7}{9}\left(\left(10-1\right)+\left({10}^2-1\right)+\left({10}^3-1\right)+....+\left({10}^n-1\right)\right]=\frac{7}{9}\left(\left(10+{10}^2+{10}^3+....+{10}^n\right)-n\right]=\frac{7}{9}\left(\frac{10\left({10}^n-1\right)}{10-1}-n\right]=\frac{7}{81}\left({10}^{n+1}-10-9n\right]$.