Let a_n = F_n+1 / F_n, where {F_n} is the Fibonacci sequence. The sequence {a_n} has a limit. We do not prove this fact, but investigate the value of the limit in these exercises.

Denote the limit of {a_n} by L. Given that the limit exists, we can determine L as follows:

(a) Show that a_n+1 = 1 + 1 a_n.

(b) Given that {a_n} converges to L, it follows that {a_n+1} also converges to L. Show that L² − L − 1 = 0 and solve this equation to determine L. (The value of L is known as the golden ratio. It arises in many different situations in mathematics.)