The Fibonacci sequence is the sequence of numbers, 0,1,1,2,3,5,8,13,21,34,55,89,..., de?ned by the base cases, F0 =0and F1 =1, and the general-case recursive de?nition, Fk = Fk?1 +Fk?2,fork ?2. Show,by induction, that, for k ?3, Fk ? ?^k?2, where ? = (1 +?5)/2 ? 1.618, which is the well-known golden ratio that traces its history to the ancient Greeks. Hint: Note that ?^2 = ?+1; hence,?^k = ?^k?1 +?^k?2, for k ?3.

Solve below question above question is the link to second question:

Show, by induction, that the minimum number, nh, of internal nodes in an AVL tree of height h, as de?ned in the proof of Theorem 4.1, satis?es the following identity,for h?1: nh = Fh+2 ?1, where Fk denotes the Fibonacci number of order k, as de?ned in the previous exercise(above question)