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Question
Express the following recurring decimal as a rational number:
`2.bar(4)`
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Solution
`2.bar(4)` = 2 + 0.4 + 0.04 + 0.004 + ... ...(1)
These terms after the first term form a G.P. whose first term is a = 0.4 and common ratio = r = 0.1
Since |r| = |0.1| = 0.1 < 1, the sum to infinity of this G.P. exists and
S = `"a"/(1 - "r")`
= `0.4/(1 - 0.1)`
= `0.4/0.9`
= `4/9`
∴ from (1), `2.bar(4) = 2 + 4/9 = 22/9`
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