Use mathematical induction to prove the statement is true for all positive integers n. = (n + 2)(n + 1)

Question

What will be an ideal response?

Answer 1

Answers may vary. Possible answer:
First we show that the statement is true when n = 1.
 For n = 1, we get   = (1 + 2)(1 + 1)
 Since  =  = 6 = 3 ? 2 = (1 + 2)(1 + 1), P₁ is true and the first condition for the principle of induction is satisfied.
Next, we assume the statement holds for some unspecified natural number k. That is,
P_k:  = (k + 2)(k + 1) is assumed true.
On the basis of the assumption that P_k is true, we need to show that P_k+1 is true. 
P_k+1:  = ((k + 1) + 2)((k + 1) + 1)
So we assume that  is true and multiply both sides of the equation by  
 ?  = (k + 2)(k + 1) ? 
 = (k + 2)(k + 3) 
 = (k + 2)(k + 3)
 = ((k + 1) + 2)((k + 1) + 1)

 The last equation says that P_k+1 is true if P_k is assumed to be true. Therefore, by the principle of mathematical induction, the statement   = (n + 2)(n + 1) is true for all natural numbers n.