Question

Find the sum to n terms :

$5+55+555+...$

• A.
  $\frac{5}{81}[{10}^{n+1}-10-9n]$
• B.
  
  $\frac{5}{77}[{10}^n-9-10n]$
• C.
  $\frac{5}{81}[9^n-10-10n]$
• D.
  None of these.

Answer 1

The correct option is A

$\frac{5}{81}[{10}^{n+1}-10-9n]$
$5+55+555+...tonterms=5(1+11+111+..tonterms)=\frac{5}{9}(9+99+999+...tonterms)=\frac{5}{9}\left[\right(10-1)+(100-1)+(1000-1)+...tonterms]=\frac{5}{9}\left[\right(10+100+1000+...tontems)-n)]=\frac{5}{9}[\frac{10({10}^n-1)}{10-1}-n]=\frac{5}{9}[\frac{{10}^{n+1}-10}{9}-n]=\frac{5}{9}\left[\frac{{10}^{n+1}-10-9n}{9}\right]=\frac{5}{81}{10}^{n+1}-10-9n)$

So, option a is correct.