14. What is wrong with the following \"proof\" of the \"fact\" that for all n E

Question

14. What is wrong with the following "proof" of the "fact" that for all n E N, the number n2 + n is odd? Proof. Let P(n) be the statement "n2 +n is odd." We will prove that P(n) is true for all n N. Suppose for induction that P(k) is true, that is, that k2 + k is odd. Now consider the statement P(k +1). Now (k 1)2 (k+1)-k2+2k +1 +k+1-k+ 2k + 2. By the inductive hypothesis, k2 +k is odd, and of course 2k2 is even. An odd plus an even is always odd, so therefore k1)2(k +1) is odd. Therefore by the principle of mathematical induction, P(n) is true for all n E N QED

Explanation / Answer

Mistake in Proof: P(1) is NOT TRUE. and proof doesn't mention this.

Whenever we use Principle of Mathematical induction, we first must check and prove a Base case. In our question, first, we have to check whether P(1) is TRUE or not i.e. 1^2 + 1 = 2 is odd or not. Clearly, 2 is not odd. So we can conclude that P(1) is FALSE. Hence, the next two steps of the induction principles, which are being described in perfectly correct manner in "Proof" part of the answer, are not allowed to be executed simply because P(1) is FALSE.

Usual logic for using mathematical induction principle is First, Prove that P(1) is TRUE. Then assume P(k) is TRUE for all k in Natural number set, using which, we have to prove that P(k+1) is TRUE. And since we have already proved P(1) is TRUE, we can conclude that P(2) is True, P(3) is TRUE and so on.

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