Use mathematical induction to prove that the statement is true for every positive integer n.2 ? 4 + 3 ? 5 + 4 ? 6 + . . . + (n + 1)(n + 3) =

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. Then, 2 ? 4 = 8 =  =  = 8. Thus, the statement is true for n = 1.
b). Assume that 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) + 1)((k + 1) + 3) = 
 Substitute the expression for S_k into the one for S_k+1:
  + ((k + 1) + 1)((k + 1) + 3) = 
 By collecting all terms on the left-hand side over a common denominator and then expanding numerators on both sides, one can show that the equality holds.

Since the equality holds, 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. Thus, it is true for all natural numbers n.