Principle of Strong Mathematical InductionLet P(n) be a statement defined for integers n; a and b be fixed integers with ab.Suppose the following statements are true:1. P(a), P(a+1), , P(b) are all true (basis step)2. For any integer k>b, if P(i) is true for all integers i with ai

*Example: Divisibility by a PrimeTheorem: For any integer n2,n is divisible by a prime. P(n)Proof (by strong mathematical induction):1) Basis step: The statement is true for n=2 P(2)because 2 | 2 and 2 is a prime number.2) Inductive step:Assume the statement is true for all i with 2i

Example: Divisibility by a PrimeProof (cont.):We have that for all iZ with 2i

Proving a Property of a SequenceProposition: Suppose a0, a1, a2, is defined as follows: a0=1, a1=2, a2=3, ak = ak-1+ak-2+ak-3 for all integers k3.Then an 2n for all integers n0. P(n)Proof (by strong induction): 1) Basis step: The statement is true for n=0: a0=1 1=20 P(0) for n=1: a1=2 2=21 P(1) for n=2: a2=3 4=22 P(2)

Proving a Property of a SequenceProof (cont.): 2) Inductive step: For any k>2,Assume P(i) is true for all i with 0i