The value of 0.mathop {234}limits^{,,, bullet | vedclass

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Question

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

Vedclass · Accepted Answer

$\frac{{232}}{{990}}$

Vedclass · Answer

(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 } ight]$

$ = \frac{2}{{10}} + 34\left[ {\frac{{1/{{10}^3}}}{{1 - 1/1000}}} ight] = \frac{2}{{10}} + 34 	imes \frac{1}{{1000}} 	imes \frac{{100}}{{99}}$

$ = \frac{2}{{10}} + \frac{{34}}{{990}} = \frac{{232}}{{990}}$.

The value of $0.\mathop {234}\limits^{\,\,\, \bullet \,\, \bullet } $ is

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