Suppose that c0, c1, c2, ... is a sequence defined as follows:
c0 = 2, c1 = 2, c2 = 6,
ck = 3ck - 3 for every integer k ≥ 3.
Prove that cn is even for each integer n ≥ 0.
Proof (by strong mathematical induction): Let the property P(n) be the following sentence.
cn is even.
We will show that P(n) is true for every integer n ≥ 0.
Show that P(0), P(1), and P(2) are true:
P(0), P(1), and P(2) are the statements "c0 is even," "c1 is even," and "c2 is even," respectively. These statements are true because 2 and 6 are even.
Show that for every integer k ≥ 2, if P(i) is true for each integer i from 0 through k, then P(k + 1) is true:
Let k be any integer with k ≥ 2, and suppose ci is even for every integer i with 0 ≤ i ≤ k. This is the inductive hypothesis.
We must show that ck + 1 is even. By definition of c0, c1, c2, ... ,
ck + 1 = 3ck - 2.
Since k ≥ 2, we have that 0 ≤ k - 2 ≤ k. So we can apply the inductive hypothesis to conclude that ck + 1 is 3 times an even integer. Thus ck + 1 is even.