8. Sequences and Series
medium

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

A

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

B

$\frac{{419}}{{999}}$

C

$\frac{{417}}{{990}}$

D

$\frac{{417}}{{999}}$

(IIT-1973)

Solution

(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}}$.

Standard 11
Mathematics

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