Question

Find the sum of the following series up to n terms:

(i) 5 + 55 + 555 + .....

(ii) .6 + .66 + .666 + .....

Answer 1

(i) Let $S_n=5+55+555+\dots \dots \text{upto n terms}$

$=5[1+11+111+\dots \dots \text{upto n terms}]$

$=\frac{5}{9}[9+99+999+\dots \text{upto n terms}]$

$=\frac{5}{9}\left[\right(10-1)+({10}^2-1)+({10}^3-1)+\dots \dots \text{upto n terms}]$

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

$=\frac{5}{9}\left[\frac{10}{9}\right({10}^n-1)-n]$

$=\frac{50}{81}({10}^n-1)-\frac{5}{9}n$

(ii) Let $S_n=.6+.66+.666+\dots \dots \text{upto n terms}$

$=6[.1+.11+.111+\dots \text{upto n terms}]$

$=\frac{6}{9}[.9+.99+.999+\dots \text{upto n terms}]$

$=\frac{2}{3}[\frac{9}{10}+\frac{99}{100}+\frac{999}{1000}+\dots \text{upto n terms}]$

$=\frac{2}{3}[(1-\frac{1}{10})+(1-\frac{1}{{10}^2})+(1-\frac{1}{{10}^3})+\dots \text{upto n terms}]$

$=\frac{2}{3}[n-(\frac{1}{10}+\frac{1}{{10}^2}+\frac{1}{{10}^3}+\dots \text{upto n terms})]$

$=\frac{2}{3}[n-\left\{\frac{\frac{1}{10}(1-\frac{1}{{10}^n})}{1-\frac{1}{10}}\right\}]$

$=\frac{2}{3}[n-\frac{1}{9}(1-\frac{1}{{10}^n})]\Rightarrow [\frac{2n}{3}-\frac{2}{27}(1-\frac{1}{{10}^n})]$