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8. Sequences and Series
medium
The value of $\overline {0.037} $ where, $\overline {.037} $ stands for the number $0.037037037........$ is
A
$\frac{{37}}{{1000}}$
B
$\frac{1}{{27}}$
C
$\frac{1}{{37}}$
D
$\frac{{37}}{{999}}$
Solution
(d) Given series $0.037037037……$
$= 0.037+0.000037+0.0000000037+…….$
= $\frac{{37}}{{{{10}^3}}} + \frac{{37}}{{{{10}^6}}} + \frac{{37}}{{{{10}^9}}} + ……$
= $37\left[ {\frac{1}{{{{10}^3}}} + \frac{1}{{{{10}^6}}} + \frac{1}{{{{10}^9}}} + ….} \right]$
= $37\left[ {\frac{{1/{{10}^3}}}{{1 – 1/{{10}^3}}}} \right] $
$= 37\left[ {\frac{1}{{{{10}^3}}}.\frac{{{{10}^3}}}{{999}}} \right]$ = $\frac{{37}}{{999}}$.
Standard 11
Mathematics
