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

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a).  Let n = 1. Then, 1² =  =  = 1. So, the statement is true for n = 1.
b). Assume the statement is true for n = k:
 S_k = .
 Also, if the statement is true for n = k + 1, then 
 S_k+1 = S_k + (3(k + 1) - 2)² = .
 Subtract to get:
 S_k+1 - S_k = (3(k + 1) - 2)² =  -  
 Expand both sides and collect like terms:
 9k² + 6k + 1 =  - 
 9k² + 6k + 1 =  = 9k² + 6k + 1
Since the equality holds, then the statement is true for n = k + 1 as long as it is true for n = k. Furthermore, the statement is true for n = 1. Therefore, the statement is true for all natural numbers n.