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2009 AMC 12A Problem 7

Problem 7 of 25EasierAlgebra

The first three terms of an arithmetic sequence are 2x−3,2x - 3, 5x−11,5x - 11, and 3x+13x + 1 respectively. The nnth term of the sequence is 2009.2009. What is n?n?

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Solution

Equal consecutive differences give (5x−11)−(2x−3)(5x - 11) - (2x - 3) =(3x+1)−(5x−11),= (3x + 1) - (5x - 11), that is 3x−8=−2x+12,3x - 8 = -2x + 12, so x=4.x = 4. The first three terms are 5,9,13,5, 9, 13, with common difference 4.4. The nnth term satisfies 2009=5+(n−1)⋅4,2009 = 5 + (n - 1)\cdot 4, so n−1=501n - 1 = 501 and n=502.n = 502. Thus, the correct answer is B.

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Concepts: arithmetic sequence

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