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in Geometric Progressions by (15.7k points)

Express 0.\(\bar 6\) as a rational number.

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1 Answer

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by (15.2k points)

Let ,x = 0.6666… 

⇒ x = 0.6 + 0.06 + 0.006 + … 

⇒ x = 6(0.1 + 0.01 + 0.001 + 0.0001 + …∞)

⇒ x = 6 \(\left(\frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \frac{1}{10000} + .... \infty\right)\)

This is an infinite geometric series. 

Here, a = 1/10 and r = 1/10

\(\therefore\) X = 6 \(\times\) 1/9 = 6/9 = 2/3

0.\(\bar 6\) = \(\frac{2}{3}\)

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