(a) $0.\mathop {234}\limits^{\,\,\, \bullet \,\, \bullet } = 0.2343434…..$                   

= $0.2 + 0.034 + 0.00034 + 0.0000034 +………….$                                         

= $0.2 + \frac{{34}}{{1000}} + \frac{{34}}{{100000}} + \frac{{34}}{{10000000}} + …..\infty $

$ = \frac{2}{{10}} + 34\left[ {\frac{1}{{{{10}^3}}} + \frac{1}{{{{10}^5}}} + \frac{1}{{{{10}^7}}} + ……..\infty } \right]$

$ = \frac{2}{{10}} + 34\left[ {\frac{{1/{{10}^3}}}{{1 – 1/1000}}} \right] = \frac{2}{{10}} + 34 \times \frac{1}{{1000}} \times \frac{{100}}{{99}}$

$ = \frac{2}{{10}} + \frac{{34}}{{990}} = \frac{{232}}{{990}}$.