Question

Discrete Mathematics

3. The sequence bo, bi, b2, is defined as follows: bo 0, bnd for integers n 22, bn- ehne (a) Calculate b2, b3, ba and bs (b) Use part (a) to guess a formula for bn for all integers n 20. c) Prove by induction on n that your guess in part (b) is correct.

Answer 1

Given sequence b_0, b_1, b_2 .....is defined as follows b_0 = 0, b_1 = 1/2 and for integers n greaterthanorequalto 2 b_n = squareroot b_n - 1 b_n - 2 + 3n/2 - 1 a) Calculate b_2, b_3, b_4 and b_5 b_2 = squareroot b_2 - 1 n_2 - 2 + 3x^2/2 - 1 = squareroot b_1 b_0 + 3 - 1 = squareroot 1/2 times 0 b_2 = 2 b_3 = squareroot b_3 - 1 b_3 - 2 + 3 times 3/2 - 1 = squareroot b_2 b_1 + 9/2 - 1 = squareroot 2 times 1/2 + 9.2 - 1 = squareroot 1 + 9/2 - 1 = 1 + 9/2 - 1 b_3 = 9/2 b_4 = squareroot b_4 - 1 b_4 - 2 + 3 times 4/2 - 1 = squareroot b_3 b_2 + 6 - 1 = squareroot 9/2 times 2 + 5 = 3 + 5 = 8 b_4 = 8 b_5 = squareroot b_4 b_3 + 3 times 5/2 - 1 = squareroot 8 times 9/2 + 15/2 - 1 Rightarrow squareroot 9 times 4 + 15/2 - 1 = 3 times 2 + 15/2 - 1 Rightarrow 6 + 15/2 - 1 Rightarrow 10 + 15/2 Rightarrow b_5 = 25/2