Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.1 + 4 + 42 + ... + 4n - 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 1 (or 4^{[(1) - 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 . That is, we need to show that
 
So we assume that  is true and add the next term, 4^k, to both sides of the equation.


Condition II is satisfied. As a result, the statement is true for all natural numbers n.