Use mathematical induction to prove the statement is true for all positive integers n.12 + 42 + 72 + . . . + (3n - 2)2 =

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 1² = 
 Since  =  = 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: 12 + 42 + 72 + . . . + (3k - 2)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: 
So we assume that  is true and add the next term,  to both sides of the equation.  
12 + 42 + 72 + . . . + (3k - 2)2 + (3(k + 1) - 2)2 =  + (3(k + 1) - 2)2
12 + 42 + 72 + . . . + (3k - 2)2 + (3(k + 1) - 2)2 =  + (9k² + 6k + 1)
12 + 42 + 72 + . . . + (3k - 2)2 + (3(k + 1) - 2)2 =  
12 + 42 + 72 + . . . + (3k - 2)2 + (3(k + 1) - 2)2 =  
12 + 42 + 72 + . . . + (3k - 2)2 + (3(k + 1) - 2)2 = 
 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    is true for all natural numbers n.