Use mathematical induction to prove the statement is true for all positive integers n.n(n + 6) < (n + 3)2

Question

What will be an ideal response?

Answer 1

Answers may vary. Possible answer:
First, we show the statement is true when n = 1.
For n = 1, 1(1 + 6) = 7 < 16 = (1 + 3)²
So 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(k + 6) < (k + 3)²  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)((k + 1) + 6) < ((k + 1) + 3)² 
(k + 1)((k + 1) + 6) = (k + 1)(k + 7) = k² + 8k + 7 = k(k + 6) + (2k + 7) 
Since P_k is assumed true, 
 =   <  =  = 
So P_k+1 is true if P_k is assumed true. Therefore, by the principle of mathematical induction,  for all natural numbers n.