Question of limits IIT JAM

Question

Let a_n=b_n+1/b_n,

where b₁=1

b₂=1

b_n+2 = b_n+1 + b_{n ,} n belongs to natural no. group

FIND Lim a_n

where n tends to infinity.

Answer 1

b_n is the famous Fibonacci sequence and you are looking for the limit of the ratio of successive terms.

The general solution to the defining difference equation is C₁pⁿ + C₂qⁿ where

p = (1/2)[1 + sqrt(5)] and q = (1/2)[1 - sqrt(5)].

As n goes to infinity qⁿ term goes to zero and the ratio of the successive terms goes to p, which is usually called the golden ratio.

There is an enormous literature about this ubiquitous sequence and if you are interested you should look up this literature. To understand how to get the general solution of the difference equation you should consult a reference on The Calculus of Finite Differences such as a book by that title by Jordan.