Question

Find the sum to 9 terms of the series $0\times 3+0.33+0.333+\cdots \cdots$

• A. $\frac{{8.10}^{10}+1}{{27.10}^9}$
• B. $\frac{{9.10}^{10}+1}{{27.10}^9}$
• C. $\frac{{9.10}^9+1}{{27.10}^8}$
• D. $\frac{{8.10}^9+1}{{27.10}^8}$

Answer 1

The correct option is A $\frac{{8.10}^{10}+1}{{27.10}^9}$

Consider the given series.

$0.3+0.33+0.333+.......$

Sum upto $9$ terms,

$S=3[0.1+0.11+0.111+...........+9terms]$

$S=\frac{3}{9}[0.9+0.99+0.999+...........+9terms]$

$S=\frac{1}{3}\left[\right(1-0.1)+(1-0.01)+(1-0.001)+...........+9terms]$

$S=\frac{1}{3}[9-(1+0.1+0.01+0.001+...........+9terms\left)\right]$

$S=\frac{1}{3}[9-\frac{0.1(1-(0.1{)}^9)}{1-0.1}]$

$S=\frac{1}{3}[9-\frac{(1-(0.1{)}^9)}{9}]$

$S=\frac{1}{3}\left[\frac{81-1+(0.1{)}^9}{9}\right]$

$S=\frac{1}{3}\left[\frac{80+{\left(\frac{1}{10}\right)}^9}{9}\right]$

$S=\frac{1}{3}\left[\frac{{80.10}^9+1}{{9.10}^9}\right]$

$S=\frac{{8.10}^{10}+1}{{27.10}^9}$

Hence, this is the answer.