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