Question 1

(a) Use Mathematical Induction to prove that for n ∈ N,

(14 marks)

(b) Hence evaluate

giving your answer correct to 3 decimal places. (6 marks)

Question 2

(a) Give a counter-example to show that the following statement is false.

(5 marks)

(b) Let a,b be positive integers. Use either the method of contraposition or the method of contradiction to show that if 12+a11+b ≤ 0.8, then a ≤ 1 or b ≥ 6.
(8 marks)

(c) Assuming the fact that √√2 is irrational, show by the method of contradiction that 6−√3 is irrational. (7 marks)

Question 3

(a) Use the Euclidean algorithm to find d = gcd(30225,2025) and express d as an integral linear combination of 30225 and 2025. (10 marks)

(b) Determine all integers d such that the equation
2025x+33y = d has integer solutions x, y ∈ Z.                  (10 marks

Question 4

(a) Employ a truth table to determine whether the following argument is valid.
                                             (10 marks)

(b) Use the rules of inference in propositional logic to show that the following argument form is valid.