The decimal expansion of 1/49

Question

Solve in your head, no long division. What is 	frac{1}{49} as a decimal, to 8 decimal places? Aim: under 20 seconds.

Accepted Answer

\frac{1}{49} = \frac{1}{50 - 1} = \frac{1/50}{1 - 1/50} = \frac{1}{50}\left(1 + \frac{1}{50} + \frac{1}{50^2} + \dotsight). Since 	frac{1}{50} = 0.02, the series stamps successive doublings two decimal places apart: 0.02 + 0.0004 + 0.000008 + 0.00000016 + \dots Read off: 	frac{1}{49} = 0.02040816\ldots, the pairs 02, 04, 08, 16 are powers of two. (The pattern continues with 32, then blurs: 64 absorbs a carry from 128 and shows up as 65.) True value: 0.0204081632\ldots Takeaway. \frac{1}{n-1} = \frac{1}{n}(1 + \frac1n + \frac1{n^2} + \cdots) turns divisions by 49, 98, 998, etc. into geomet…

The decimal expansion of 1/49

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