HMMT problem
Number Theory
Level
4
What is the sum of all unit fractions whose decimal expansion is terminating?
2
2.5
e
3
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1 solution
A decimal expansion of a unit fraction terminates if and only if the denominator is of the form (2^n)×(5^m) for non-negative integers n and m. Thus, the sum is: (1 + 1/2 + 1/4 + ... ) + 1/5×(1 + 1/2 + 1/4 + ...) + 1/5²×(1 + 1/2 + 1/4 + ...) + ... = 2×5/4 = 2.5 .