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Question
Express the following recurring decimal as a rational number:
`2.3bar(5)`
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Solution
`2.3bar(5)` = 2.3555 ...
= 2.3 + 0.05 + 0.005 + 0.0005 + …
The terms 0.05, 0.005, 0.0005 are in G.P.
∴ a = 0.05, r = `0.005/0.05` = 0.1
Since |r| = |0.1| < 1
∴ Sum to infinity exists.
∴ Sum to infinity = `2.3 + "a"/(1 - "r")`
= `2.3 + 0.05/(1 - 0.1)`
= `2.3 + 0.05/0.9`
= `23/10 + 5/90`
= `212/90`
= `106/45`
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