2004 AMC 12B Problem 12

Below is the professionally curated solution for Problem 12 of the 2004 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12B solutions, or check the answer key.

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Concepts:recursion arithmetic sequence pattern recognition

Difficulty rating: 1500

12.

In the sequence $2001, 2002, 2003, \ldots,$ each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001 + 2002 - 2003 = 2000.$ What is the $2004$ th term in this sequence?

$-2004$

$-2$

$0$

$4003$

$6007$

Solution:

The rule gives $2001, 2002, 2003, 2000, 2005, 1998, \ldots$ The even-indexed terms are $2002, 2000, 1998, \ldots,$ decreasing by $2.$ The $2004$ th term is the $1002$ nd of these: $2002 + 1001(-2) = 0.$

Thus, the correct answer is C.

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