Use mathematical induction to prove that the statement is true for every positive integer n.1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + n(n + 1) =

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. Then, 1?2 = 2 =   = 2. 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 + (k + 1)(k + 2) = .
 Subtracting, we get:
 S_k+1 - S_k = (k + 1)(k + 2) =  - .
 Expand both sides and collect like terms:
 k² + 3k + 2 =  -  =  = k² + 3k + 2
 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.