Most people look at the first four digits of 1/7 ~= .1428 and think that the fact that 7*2=14 and 7*4=28 is a mere coincidence. It’s not. Let’s compute the decimal expansion of 1/7 without doing long division, but using instead $$x/(1-x) = x + x^2 + x^3 + \ldots.$$

1/7

= 7/49

= 7*1/(50-1)

= 7*1/50/(1-1/50)

= 7*.02/(1-.02)

= 7*(.02 + .02^2 + .02^3 + …)

= 7*(.02 + .0004 + .000008 + .00000016 + .0000000032 + …)

= 7*(.020408163264…)

= 7*(.020408) + 7*(..000000163264)

= .142856 + 7*(.00000016) + 7*(.00000000326…)

= .142856 + .00000112 + 7*(.00000000326…)

Notice that the first n digits must be .142857. The decimal expansion of 1/j either terminates or repeats with at most (j-1) repeating digits and we have 6 digits, so those six digits must repeat, thus

1/7 = .142857142857142857142857142857142857….

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