Use mathematical induction to prove that the statement is true for every positive integer n.4 ? 6 + 5 ? 7 + 6 ? 8 + . . . +(n + 3)(n + 5) = 
What will be an ideal response?
Answers will vary. One possible proof follows.
a). Let n = 1. Then 4 ? 6 = 24 = =
= 24. Thus, the statement is true for n = 1.
b). Assume that the statement is true for n = k:
Sk =
Also, if the statement is true for n = k + 1, then
Sk+1 = Sk + ((k + 1) + 3)((k + 1) + 5) =
Subtract to get:
Sk+1 - Sk = ((k + 1) + 3)((k + 1) + 5) = -
Expand both sides and collect like terms:
k2 + 10k + 24 = -
=
k2 + 10k + 24 = k2 + 10k + 24
Since the equality holds, then the statement is true for n = k + 1 as long as it is true for n = k. Furthermore, it is true for n = 1. Therefore, the statement is true for all natural numbers n.
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