Question

Find the sum to n terms of the series
7 + 77 + 777 + 7777 + ...

• A. $\frac{7}{9}\left\{10\right({10}^n-1)-n\}$
• B. $\frac{7}{9}\left\{\frac{10}{9}\right({10}^n-1)-1\}$
• C. $7\left\{\frac{10}{9}\right({10}^n-1)-1\}$
• D. none of these

Answer 1

The correct option is B $\frac{7}{9}\left\{\frac{10}{9}\right({10}^n-1)-1\}$

The best way is to go through options.
Consider n = 2, then $S_2=7+77=84$
Now, put n = 2 in choice (b).

Conventional Approach:
$S_n=7+77+777+...$ to n terms
$=7(1+11+111+...$ to n terms)
$=\frac{7}{9}$ (9 + 99 + 999+...to n terms)
$=\frac{7}{9}$ {(10-1) + (100-1} + (1000-1)+... to n terms}
$=\frac{7}{9}$ {(10+100+1000+... to n terms) - (1 + 1 +1...to n terms)}
$=\frac{7}{9}\{\frac{10.({10}^n-1)}{10-1}-n\}=\frac{7}{9}\left\{\frac{10}{9}\right({10}^n-1)-n\}$