Question

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?

$-2004$

$-2$

$0$

$4003$

$6007$

Answer: C

Concepts:recursion arithmetic sequence pattern recognition

Difficulty rating: 1500

Solution:

The rule gives $2001,$ $2002,$ $2003,$ $2000,$ $2005,$ $1998, \ldots$ The even-indexed terms are $2002, 2000, 1998, \ldots,$ decreasing by $2.$

More precisely, the recurrence verifies inductively that $a_{2k} = 2004 - 2k$ and $a_{2k+1} = 2001 + 2k.$ Therefore $\begin{aligned} a_{2004} &= a_{2\cdot1002} \\ &= 2004 - 2(1002) = 0. \end{aligned}$

Thus, the correct answer is C.

2004 AMC 12B Problem 12

Answer: C

Difficulty rating: 1500

Solution:

Problem 12 in Other Years