0.mathop {423}limits^{,,,,, bullet , bullet , | vedclass

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Question

(a) We have $0.\mathop {423}\limits^{\,\,\,\,\, \bullet \, \bullet \,} = 0.4232323.......$

$ = 0.4 + 0.023 + 0.00023 + 0.0000023 + ........\in

Vedclass · Accepted Answer

$\frac{{419}}{{990}}$

Vedclass · Answer

(a) We have $0.\mathop {423}\limits^{\,\,\,\,\, \bullet \, \bullet \,} = 0.4232323.......$

$ = 0.4 + 0.023 + 0.00023 + 0.0000023 + ........\infty $

$ = \frac{4}{{10}} + \frac{{23}}{{{{10}^3}}} + \frac{{23}}{{{{10}^5}}} + \frac{{23}}{{{{10}^7}}}........\infty $

$ = \frac{4}{{10}} + \frac{{23}}{{{{10}^3}}}\left[ {1 + \frac{1}{{{{10}^2}}} + \frac{1}{{{{10}^4}}} + ..........\infty } \right]$

$ = \frac{4}{{10}} + \frac{{23}}{{1000}}\left( {\frac{1}{{1 - \frac{1}{{{{10}^2}}}}}} \right) = \frac{4}{{10}} + \frac{{23}}{{990}} = \frac{{419}}{{990}}$.

$0.\mathop {423}\limits^{\,\,\,\,\, \bullet \, \bullet \,} = $

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