Use mathematical induction to prove the following.1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + n(n + 1) =

Question

What will be an ideal response?

Answer 1

Answers may vary. One possibility:
S_n: 1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + n(n + 1) = 
S₁: 1 ? 2 = 
S_k: 1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) = 
S_k+1: 1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) + (k + 1)(k + 2) = 
1. Basis step: Since  =  = 1 ? 2, S₁ is true.
2. Induction step: Let k be any natural number. Assume S_k. Deduce S_k+1.
1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) = 
1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) + (k + 1)(k + 2) =  + (k + 1)(k + 2)
 =  + 
 = 
 = .