Kind of easy... maybe

Calculus Level 2

Next one up is harder!

What is the average of all the digits of 2 7 \frac{2}{7} ?

(For the pesky know-it-alls: "What is the limit of the average of the first nth digits of 2 7 \frac{2}{7} , where n goes to infinity?")


The answer is 4.5.

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1 solution

Eric Hernandez
Aug 4, 2014

2 7 \frac{2}{7} repeats the digits 2, 8, 5, 7, 1, and 4, forever. And if you take the first 6 n 6n decimal places and average them (for n n an integer), you will get the answer for infinity: ( 2 + 8 + 5 + 7 + 1 + 4 ) n 6 n = 2 + 8 + 5 + 7 + 1 + 4 6 = 4.5 \frac{(2+8+5+7+1+4)n}{6n}=\frac{2+8+5+7+1+4}{6}=4.5

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