Given an example of a statement P(n) such that it is true for all n ϵ N.

If P(n) is the statement “2ⁿ ≥ 3n”, and if P(r) is true, prove that P(r + 1) is true.

Given. P(n) = “2ⁿ ≥ 3n” and p(r) is true.

Prove. P(r + 1) is true

If P(n) is the statement “n² + n” is even”, and if P(r) is true, then P(r + 1) is true

Given. P(n) = n² + n is even and P(r) is true.

Prove. P(r + 1) is true

If P(n) is the statement “n² – n + 41 is prime”, prove that P(1), P(2) and P(3) are true. Prove also that P(41) is not true.

Given. P(n) = n² - n + 41 is prime

Prove. P(1),P(2) and P(3) are true and P(41) is not true.

Prove the following by the principle of mathematical induction: