Question 93
For a given positive integer $k$, define two sequences $\{a_n\}$ and $\{b_n\}$ as follows:
$$
\begin{aligned}
a_1 &= k, & a_2 &= k, & a_{n+2} &= a_n a_{n+1} \quad (n \ge 1) \\
b_1 &= 1, & b_2 &= k, & b_{n+2} &= \frac{b_{n+1}^3 + 1}{b_n} \quad (n \ge 1)
\end{aligned}
$$
For any positive integer $n$, show that $a_{2n}b_{n+3}$ is an integer.
$$
\begin{aligned}
a_1 &= k, & a_2 &= k, & a_{n+2} &= a_n a_{n+1} \quad (n \ge 1) \\
b_1 &= 1, & b_2 &= k, & b_{n+2} &= \frac{b_{n+1}^3 + 1}{b_n} \quad (n \ge 1)
\end{aligned}
$$
For any positive integer $n$, show that $a_{2n}b_{n+3}$ is an integer.