Question 88
Given positive integers $n$, $a_1$, $a_2$, $\dots$, $a_n$, define $q_0 = 1$, $q_1 = a_1$ and $q_{k+1} = a_{k+1}q_k + q_{k-1}$, for $1 \le k \le n-1$.
Prove that, given $c \gt 1$, there exists $K \gt 0$ such that, for all $M \gt K$, there exist a positive integer $n$ and $a_1, a_2, \dots, a_n \in \{1, 2\}$ such that $M \le q_n \lt c \cdot M$.
Prove that, given $c \gt 1$, there exists $K \gt 0$ such that, for all $M \gt K$, there exist a positive integer $n$ and $a_1, a_2, \dots, a_n \in \{1, 2\}$ such that $M \le q_n \lt c \cdot M$.