Question 115
Let $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ be sequences of real numbers such that $a_0 = 40$, $b_0 = 41$ and for all $n \ge 0$ we have
$$
a_{n+1} = a_n + \frac{1}{b_n} \quad \text{and} \quad b_{n+1} = b_n + \frac{1}{a_n}
$$
Find the smallest positive integer $k$ such that $a_k \gt 80$.
$$
a_{n+1} = a_n + \frac{1}{b_n} \quad \text{and} \quad b_{n+1} = b_n + \frac{1}{a_n}
$$
Find the smallest positive integer $k$ such that $a_k \gt 80$.