Use mathematical induction to prove the following.12 + 42 + 72 + . . . + (3n - 2)2 =

Question

What will be an ideal response?

Answer 1

Answers may vary. One possibility:
S_n: 12 + 42 + 72 + . . . + (3n - 2)2 = 
S₁: 12 = 
S_k: 12 + 42 + 72 + . . . + (3k - 2)2 = 
S_k+1: 12 + 42 + 72 + . . . + (3k - 2)2 + [3(k + 1) - 2]2 = 
1. Basis step: Since  =  =  = 1 = 12, S₁ is true.
2. Induction step: Let k be any natural number. Assume S_k. Deduce S_k+1.
12 + 42 + 72 + . . . + (3k - 2)2 = 
12 + 42 + 72 + . . . + (3k - 2)2 + [3(k + 1) - 2]2 =  + [3(k + 1) - 2]2
 =  + (3k + 1)2
 =  + (9k2 + 6k + 1)
 =  + 
 = 
 = 
 = 
 = 
 = .