These questions are from my tutorial session but didn't understand well! Consider the sequence dened recursively by s0 = 0, s1 = 1, s2 = 0; sn := sn3...

These questions are from my tutorial session but didn't understand well! need some help

4. Consider the sequence defined recursively by s0 = 0, s1 = 1, s2 = 0; sn := sn−3 + sn−2 −sn−1 for n ≥ 3 .

(a) Calculate s3, s4 and s5 and make a guess as to the general form of sn.

(b) Use induction to prove that your guess is correct.

8. For a positive integer n, let p(n) be the statement "n2 −3n is odd".

(a) Show that p(n) ⇒ p(n + 1) for each positive integer n.

(b) For which numbers n is p(n) true?

9. Let T(n) = T(n−1) + 2T(b n 2c) for n ≥ 1, let T(0) = 1, (where b·c is the floor function).

(a) Evaluate T(1), T(2) and T(3).

(b) Use induction to prove that T(n) is divisible by 3 for all n ≥ 1.