2009 AMC 12A Problem 7

Below is the professionally curated solution for Problem 7 of the 2009 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12A solutions, or check the answer key.

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Concepts:arithmetic sequence

Difficulty rating: 1250

The first three terms of an arithmetic sequence are $2x - 3,$ $5x - 11,$ and $3x + 1$ respectively. The $n$ th term of the sequence is $2009.$ What is $n?$

$255$

$502$

$1004$

$1506$

$8037$

Solution:

Equal consecutive differences give $(5x - 11) - (2x - 3) = (3x + 1) - (5x - 11),$ that is $3x - 8 = -2x + 12,$ so $x = 4.$

The first three terms are $5, 9, 13,$ with common difference $4.$

The $n$ th term satisfies $2009 = 5 + (n - 1)\cdot 4,$ so $n - 1 = 501$ and $n = 502.$

Thus, the correct answer is B.

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