2002 AIME I Problems/Problem 7
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers and with ,
What are the first three digits to the right of the decimal point in the decimal representation of ?
will always be 1, so we can ignore those terms, and using the definition ():
Since the exponent of the goes down extremely fast, it suffices to consider the first few terms. Also, the term will not affect the digits after the decimal, so we need to find the first three digits after the decimal in
(The remainder after this term is positive by the Remainder Estimation Theorem). Since the repeating decimal of repeats every 6 digits, we can cut out a lot of 6's from to reduce the problem to finding the first three digits after the decimal of
That is the same as , and the first three digits after are .
An equivalent statement is to note that we are looking for , where is the fractional part of a number. By Fermat's Little Theorem, , so ; in other words, leaves a residue of after division by . Then the desired answer is the first three decimal places after , which are .
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