Question

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?

$29$

$55$

$85$

$133$

$250$

Answer: E

Concepts:recursion digits pattern recognition

Difficulty rating: 1370

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.

2005 AMC 10B Problem 11

Answer: E

Difficulty rating: 1370

Solution:

Problem 11 in Other Years