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) =

Question

What will be an ideal response?

Answer 1

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:
 S_k = 
 Also, if the statement is true for n = k + 1, then
 S_k+1 = S_k + ((k + 1) + 3)((k + 1) + 5) = 
 Subtract to get:
 S_k+1 - S_k  = ((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.