Question

Find the sum of the following series:
5+55+555+.........................to n terms​

Answer 1

To find :

5+55+555+...................to n terms.

Formula used :

Summation of G.P i.e

$Sn = a( \frac{ {r}^{n} - 1}{r - 1} )$

Solution :

5+55+555+..........n terms

$= > 5(1 + 11 + 111....n \: terms \: )$

$= > \frac{5}{9} (9 + 99 + 999 + .......... + to \: n \: terms)$

$= > \frac{5}{9} (10 - 1)( {10}^{2} - 1)( {10}^{3} - 1) + ..........( {10}^{n} - 1)$

$= > \frac{5}{9} (10 + {10}^{2} + {10}^{3} + ...... {10}^{n} ) - (1 + 1 + 1 + 1......n \: times \: )$

$= > \frac{5}{9} (10 \times \frac{ {10}^{n} - 1}{10 - 1} )$

$= > \frac{5}{9} ( \frac{10}{9} ( {10}^{n} - 1) - n)$

$= > \frac{5}{18} ( {10}^{n + 1} - 10 - 9n)$