Suppose there is a given statement involving the natural number n such that
The statement is true for n = 1, i.e., is true, and
If the statement is true for n = k (where k is some positive integer), then the statement is also true for , i.e., truth of implies the truth of Then, is true for all natural numbers n.
Prove that for all positive integers n.
When Hence P(1) is true.
Assume that is true for any positive integers k, i.e. ... (1)
We shall now prove that is true whenever is true.
Multiplying both sides of (1) by 2, we get 2. i.e.,
Therefore, is true when is true. Hence, by principle of mathematical induction, is true for every positive integer n.
For every positive integer n, prove that is divisible by 4.
: is divisible by 4.
We note that
: 71 – 31 = 4 which is divisible by 4. Thus is true for n = 1
Let be true for some natural number k,
i.e,: is divisible by 4.
We can write = 4d, where
Now, we wish to prove that is true whenever is true.
From the last line, we see that is divisible by 4. Thus, is true when P(k) is true. Therefore, by principle of mathematical induction the statement is true for every positive integer n.