2004 AMC 12B Problem 12

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

Small hint:

Write out several terms to spot a pattern

Big hint:

The even-indexed terms

2002, 2000, 1998, \ldots

form an arithmetic progression

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.