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 = 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) + 3)((k + 1) + 5) = 
 Substitute the expression for S_k into the one for S_k+1:
  + ((k + 1) + 3)((k + 1) + 5) = 
 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.