(a) Use a direct proof to show that the product of two odd numbers is odd....

(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. Maximum score = 25 (parts (a) 9 points, part (b-i) and (b-ii) 8 points)

(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 +...+ (3n – 4) = (n/2) (3n-5)

ii) ½ + ¼ + 1/8 + … + 1/2n = (2n – 1)/ 2n

Q.3. Maximum score = 25 Prove that 1 + (1/4) + (1/9) +……(1/n2) < 2 - (1/n) for n, a positive integer >1.

Q. 4 Maximum score 25 Determine the formular for an given by the recurrence relation an = an-1 + 6an -2 ; a0 = 1, a1 = 8.

Expert Solution

Colby Messinger answered 2 years ago

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