The following questions deal with proofs and recurrence relations. Thanks in advance for you help. Please show step by step solutions.

Q. 1 (a) Use a direct proof to show that the product of two odd numbers is odd (b) Prove that there are no solutions in integers x and y to the equation 2x2 5y2 14 (c) Prove that the square of an even number is an even number using (a) direct proof, (b) an indirect proof, and (c) a proof by contradiction. Q. 2 (a) Show that 13 23 +n3 n(n+1)/2]2 whenever n is a positive integer (b) Use induction to prove the following for all natural numbers n i) 1 2 5 8 (3 4) E (n/2) (3n-5) ii) 1/8 1/2n (2n l)/ 2 Q.3 Prove that l (1/4) (1/9) +......(l20 2 (1) for n, a positive integer l [Hint: Suppose the left side POn). Calculate POn-t-1). It would be 2- (n(n+1) +1) [n(n+1)2] which is