Question

1. Use mathematical induction to show that, ∀n ≥ 3, 2n² + 1 ≥ 5n

2. Letting s₁ = 0, find a recursive formula for the sequence 0, 1, 3, 7, 15,...

3. Evaluate. (a) 55mod 7. (b) −101 div 3.

4. Prove that the sum of two consecutive odd integers is divisible by 4

5. Show that if a|b then −a|b.

6. Prove or disprove: For any integers a,b, c, if a ∤ b and b ∤ c, then a ∤ c.

7. Use mathematical induction to show that, ∀n ≥ 0, 2|(3ⁿ +1)

8. Solve the following. (a) List the first four terms of the recursive sequence defined by s₁ = 1 and ∀n ≥ 2, s_n = (s_n−1 + 1) ² . (b) Given that ∑_i=1ⁿ   (i= n(n+1)/2) , find the sum 2 + 4 + 6 +...+ 200

Answer 1