Question

Let $a_0 = 2,$ $a_1 = 5,$ and $a_2 = 8,$ and for $n \gt 2$ define $a_n$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11.$ Find $a_{2018} \cdot a_{2020} \cdot a_{2022}.$

Answer

Answer: 112

Concepts:recursion modular arithmetic pattern recognition

Difficulty rating: 1970

Solution:

Computing successive terms gives $\begin{gathered} 2,\ 5,\ 8,\ 5,\ 6,\ 10,\ 7,\ 4,\ 7,\ 6,\ \\ 2,\ 5,\ 8,\ \ldots \end{gathered}$ Since $(a_{10}, a_{11}, a_{12})$ $= (2, 5, 8)$ $= (a_0, a_1, a_2)$ and each term depends only on the previous three, the sequence is periodic with period $10.$

Therefore $a_{2018} = a_8 = 7,$ $a_{2020} = a_0 = 2,$ and $a_{2022} = a_2 = 8,$ so the product is $7 \cdot 2 \cdot 8 = 112.$

2018 AIME II Problem 2

Answer: 112

Difficulty rating: 1970

Solution:

Problem 2 in Other Years