Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.12 + 42 + 72 + . . . + (3n - 2)2 =

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 1 =  = 1.
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.
12 + 42 + 72 + . . . + (3k - 2)2 + (3(k + 1) - 2)2 =  + (3(k + 1) - 2)2
 =  + (3k + 1)2
 =  + 
 = 
 = 
Simplify the expression  to verify:
   = 
 = 
 = 
 = 
Condition II is satisfied. As a result, the statement is true for all natural numbers n.