2005 AMC 10B Problem 11

Problem 11 of 25IntermediateAlgebra Number Theory

The first term of a sequence is

2005.

Each succeeding term is the sum of the cubes of the digits of the previous term. What is the

2005

th term of the sequence?

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Solution

The sequence begins

2005, 133, 55, 250, 133, \ldots,

so after the first term it repeats the cycle

133, 55, 250

of length

3.

Terms

2,

3,

and

4

are the first, second, and third entries of this cycle. Because

2005 - 2 = 2003

leaves remainder

2

upon division by

3,

the

2005

th term matches the third entry,

250.

Thus, E is the correct answer.