The decimal expansion of 1/51

Question

Solve in your head, no long division. What is 	frac{1}{51} as a decimal, to 8 decimal places? Aim: under 25 seconds.

Accepted Answer

\frac{1}{51} = \frac{1}{50 + 1} = \frac{1/50}{1 + 1/50} = \frac{1}{50}\left(1 - \frac{1}{50} + \frac{1}{50^2} - \cdotsight). Since 	frac{1}{50} = 0.02, the terms are the powers of two, two places apart, with alternating signs: 0.02 - 0.0004 + 0.000008 - 0.00000016 + \cdots Accumulate: 0.02 - 0.0004 = 0.0196; +0.000008 = 0.019608; -0.00000016 = 0.01960784; and so on. The pairs 02, 04, 08, 16 are powers of two again, now added and subtracted in turn: \frac{1}{51} = 0.01960784\ldots True value: 0.019607843\ldots The sign flip is the only change from the 	frac{1}{49} story: a denominator of n…

The decimal expansion of 1/51

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