Question 165
Let $N^*$ be the set of positive integers. Define $a_1 = 2$, and for $n = 1, 2, \dots$,
$$
a_{n+1} = \min\{\lambda \mid \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} + \frac{1}{\lambda} \lt 1, \lambda \in N^*\}.
$$
Prove that $a_{n+1} = a_n^2 - a_n + 1$ for $n = 1, 2, \dots$.
$$
a_{n+1} = \min\{\lambda \mid \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} + \frac{1}{\lambda} \lt 1, \lambda \in N^*\}.
$$
Prove that $a_{n+1} = a_n^2 - a_n + 1$ for $n = 1, 2, \dots$.