Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + n(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 2 =  = 2.
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.
1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) + (k + 1)(k + 2) =  + (k + 1)(k + 2)
 =  + 
 = 
 = 
Condition II is satisfied. As a result, the statement is true for all natural numbers n.