Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.4 + 8 + 12 + ... + 4n = 2n(n + 1)

Question

What will be an ideal response?

Answer 1

First we show that the statement is true when n = 1.
 For n = 1, we get 4 = 2(1)(1 + 1) = 4.
This is a true statement and Condition I is satisfied.

Next, we assume the statement holds for some k. That is,
  is true for some positive integer k.
We need to show that the statement holds for k + 1. That is, we need to show that
 
So we assume that  is true and add the next term,  to both sides of the equation.
4 + 8 + 12 + ... + 4k + 4(k + 1) = 2k(k + 1) + 4(k + 1)
 = (k + 1)(2k + 4)
 = 2(k + 1)(k + 2)
Condition II is satisfied. As a result, the statement is true for all natural numbers n.