Inventory021:  Probability about a sequence (proposed by Roger Cuculière, 2/11/03).

Let  u_{n}   be a real sequence defined by : 0<=u_{0}<=1, 0<=u_{1}<=1, and for n>=2 : u_{n}=(sqrt(u_{n-1})+sqrt(u_{n-2}))/2.
Prove that  the sequence (u_{n}), n>=4, is non-discreasing.
If  u_{0} and  u_{1} are choosen at random, uniformely and independantly in [0,1], find the probability that the sequence  (u_{n}) is non-discreasing respectively  for : n>=0, n>=1, n>=2, n>=3.

Bibliography.

Return to [Polya Home Page] [Members] [Problem Center] [Inventory] [Reserve Area]