Alg8.17

Find the sum of the $n\,$ terms of the series $0.5+0.55+0.555+.......n\,$ terms.

$S_n=\frac{5}{10}+\frac{55}{100}+\frac{555}{1000}+.......+n\,$ terms

$5[\frac{1}{10}+\frac{11}{100}+........n]\,$

Multiplying and dividing by 9, we get

$\frac{5}{9}[\frac{9}{10}+\frac{99}{100}+........+n]\,$

$\frac{5}{9}[(1-\frac{1}{10})+(1-\frac{1}{100})+........+n]\,$

$\frac{5}{9}[(1+1+......n)-(\frac{1}{10}+\frac{1}{100}+........)]\,$

$\frac{5}{9}[n-\frac{1}{10}\frac{1-\frac{1}{10^n}}{1-\frac{1}{10}}]\,$

$\frac{5}{9}[n-\frac{10^n-1}{9\cdot 10^n}]=\frac{5}{9}[n-\frac{10^n-1}{9\cdot 10^n}]\,$