Question

Assume we need to prove that for some statement:

P: ∀n ∈N P(n) is even.
We calculate and check: P(1) is odd. P(2) is even. P(3) is odd P(4) is odd.
What is the next step if we need to use the proof by mathematical induction?

Answer 1

The proof by mathematical induction is not applicable since the base case does not hold consistently, indicating the initial statement that P(n) is even for all natural numbers is false.

Step-by-step explanation:

To answer the student's question regarding the proof by mathematical induction where the statement P: ∀n ∈N P(n) is even is tested, we have already observed that P(1) is odd, P(2) is even, P(3) is odd, and P(4) is odd. The next step in the proof by induction would typically be to assume that P(k) holds for some arbitrary positive integer k, and then show that P(k+1) also holds. However, since the initial checks reveal a mix of odd and even outcomes, it indicates that the statement does not hold for all natural numbers. Therefore, a traditional proof by induction is not the correct method here, as the base case does not hold consistently. Instead, we should conclude that the initial statement P(n) is even for all n is false.