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 ${2005}^{\text{th}}$ term of the sequence?
I need to find a pattern or a formula for this sequence. I start by computing the first few terms: $2005, 133, 55, 250, 133, 55, \dots$. I notice that the sequence repeats after $250$, so it has a cycle of length $3$. That means that every third term is $250$, and the terms between them are $133$ and $55$. To find the ${2005}^{\text{th}}$ term, I need to divide $2005$ by $3$ and see what the remainder is. I can use long division or a calculator to do this. I get $2005 = 3 \times 668 + 1$. That means that the ${2005}^{\text{th}}$ term is the same as the first term of the cycle, which is $250$. 