Use mathematical induction to prove that the statement is true for every positive integer n.4 + 8 + 12 + . . . + 4n = 2n(n + 1)

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). If n = 1, then 4 = 2(1)(1 + 1) = 4. So, the statement is true for n = 1.
b). Assume that the statement is true for n = k:
 4 + 8 + . . . + 4k = 2k(k + 1).
 Add 4(k + 1) to both sides to obtain:
 4 + 8 + . . . + 4k + 4(k + 1) = 2k(k + 1) + 4(k + 1)
 Factor the right hand side to get:
 4 + 8 + . . . + 4k + 4(k + 1) = 2(k + 1)(k + 2)
The statement is true for n = k + 1 if it is true for n = k. Furthermore, the statement is true for n = 1. Therefore, the statement is true for all natural numbers n.