2005 AMC 12B Problem 10

Problem 10 of 25EasierAlgebra 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

since

2^3 + 0^3 + 0^3 + 5^3 = 133,

1^3 + 3^3 + 3^3 = 55,

5^3 + 5^3 = 250,

and

2^3 + 5^3 + 0^3 = 133.

After the initial

2005,

the terms cycle through

133, 55, 250

with period

3.

Term

n

for

n \ge 2

is the

((n-2)\bmod 3)

th entry of

133, 55, 250.

Since

2005 - 2 = 2003 \equiv 2 \pmod 3,

the

2005

th term is

250.

Thus, the correct answer is E.