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

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. Then, 4 =  =  = 4. Thus, the statement is true for n = 1.
b). Assume 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 + 4(k + 1) = 
 Subtract to obtain:
 S_k+1 - S_k  = 4(k + 1) =   - 
 Expand both sides and collect like terms:
 4k + 4 =  -  =  = 4k + 4
Since the equality holds, then 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. Therefore, the statement is true for all natural numbers n.