Problem - 3264
A sequence satisfies $a_1 = 3, a_2 = 5$, and $a_{n+2} = a_{n+1} - a_n$ for $n \ge 1$. What is the value of $a_{2018}$?
This particular problem can be solved by looking for patterns: \begin{align} a_1&=3\\ a_2&=5\\ a_3&=2\\ a_4&=-3\\ a_5&=-5\\ a_6&=-2\\ a_7&=3\\ a_8&=5 \end{align} Therefore, the sequence has a cycle of $6$ which implies $a_{2018}=a_{2}=\boxed{5}$. Note: a general approach to solve linear regression, i.e. $a_{n+2}=p\cdot a_{n+1} + q\cdot a_n$ where $p$ and $q$ are constant is discussed in the book %%HREF%%Competition Algebra%%Home/books/110-competition-algebra%%.