Use mathematical induction to prove the following.1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + n(n + 1) = 
What will be an ideal response?
Answers may vary. One possibility:
Sn: 1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + n(n + 1) = 
S1: 1 ? 2 = 
Sk: 1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) = 
Sk+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 Sk. Deduce Sk+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)
= 
+ 
= 
= 
.
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