Express the following recurring decimals as rational numbers. 0 .… - Vidyadip

Question

Express the following recurring decimals as rational numbers.

$0 . \overline{7}$

✓

Answer

$0 . \overline{7}=0.7777 \ldots=0.7+0.07+0.007+\ldots$

The terms 0.7, 0.07, 0.007,… are in G.P.

$\therefore a=0.7, r=\frac{0.07}{0.7}=0.1,|r|=|0.1|<1$

∴ Sum to infinity exists.

∴ Sum to infinity

$\begin{aligned}=\frac{a}{1-r} & =\frac{0.7}{1-0.1} \\ & =\frac{0.7}{0.9} \\ & =\frac{7}{9}\end{aligned}$

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