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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