2004 AMC 10B Problem 19

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: 1460

Small hint:

Write out several terms:

2001,

2002,

2003,

2000,

2005,

1998,

\ldots

Big hint:

The even-position terms decrease by

2

each time

Solution:

The recurrence

a_{k+1} = a_{k-2} + a_{k-1} - a_k

gives

a_{k+1} - a_{k-1} = -(a_k - a_{k-2}).

The sequence begins

2001,

2002,

2003,

2000,

2005,

1998,

\ldots

So the even-position terms form the arithmetic sequence

2002, 2000, 1998, \ldots

with common difference

-2.

The

2004

th term is its

1002

nd term,

2002 + 1001(-2) = 0.

Thus, the correct answer is C.