Question 138
A sequence of positive real numbers $a_1, a_2, a_3, \dots$ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, \dots$ is defined by equations $b_1 = a_1$, $b_n = a_n + (b_1 + b_3 + \dots + b_{n-1})$ for even $n \gt 1$, $b_n = a_n + (b_2 + b_4 + \dots + b_{n-1})$ for odd $n \gt 1$. Prove that if $n \ge 3$, then $\frac{1}{3} \lt \frac{b_n}{n \cdot a_n} \lt 1$.