(e) In an inductive proof that for every positive integer n, n(n+1)(2n+1) = -1 what must be proven in the inductive step? What would be the inductive hypothesis in the inductive step from your previous answer? (g) Prove by induction that for any positive integer n. win +11(2n-1) Σ²² G 7-1

Use the definitions of even, odd, prime, and compositive numbers to justify your answers for (a)-(c). Assume that r and s are particular integers. (a) Is Srs even? O Yes, because 8rs = 2(4rs) + 1 and 4rs is an integer. O Yes, because 8rs = 2(4rs) and 4rs is an integer. O No, because 8rs = 2(4rs) and 4rs is an integer. O No, because 8rs = 2(4rs) + 1 and 4rs is an integer. (b) Is 2r + 4s² + 9 odd? O Yes, because 2r + 4s² + 9 = 2(r + 2s² + 4) + 1 and r + 2s² + 4 is an integer. O Yes, because 2r + 4s² + 9 = 2(r + 2s² + 4) and r + 2s² + 4 is an integer. O No, because 2r + 4s² + 9 = 2(r + 2s² + 4) + 1 and r + 2s² + 4 is an integer. O No, because 2r + 4s² + 9 = 2(r + 2s² + 4) and r + 2s² + 4 is an integer. (c) If rand s are both positive, is r² + 2rs + s² composite? O Yes, because 2² + 2rs + s² = (r + s)² and r + s is an integer. O Yes, because r² + 2rs + s² = (r + s)2 and r + s is not an integer. O No, because 2² + 2rs + s² = (r + s)² and r + s is not an integer. O No, because 2 + 2rs + s² =…

1. Induction and prime factorisation of natural numbers: Let n E N be a natural number. A prime factorisation of n is an equation n = prime number and d; e {0}UN. Using Mathematical Induction, prove that every natural number has a prime factorisation. di d2 P1 P2 de ..Pk where for i = 1, . . . , k, The following statements inay be useful in your proof: (Q1) Vm, n E N (m 2) →

Discrete Mathematics