Talk:2006 AMC 12A Problems/Problem 24

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Define such that . Then the expression in the problem becomes: . Expanding this using binomial theorem gives (we may omit the coefficients, as we are seeking for the number of terms, not the terms themselves). Simplifying gives: . We can also take out all the 2 and all the x terms, as they will not affect the answer. Thus, we must find the number of terms in this expression: . Because , will have n+1 terms, by the binomial theorem. Thus, the expression will have terms. We can easily find this sum by noting that this is equal to the sum of the first 1004 consecutive odd integers, or , which gives us , or D.

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