MathNet 1962 — Part 01 Question 10

Given integers $a_0$, $a_1$, ..., $a_{100}$, satisfying $a_1 \gt a_0$, $a_1 \gt 0$, and $a_{r + 2} = 3 a_{r + 1} - 2 a_r$ for $r = 0$, $1$, ..., $98$. Prove $a_{100} \gt 2^{99}$.