Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.9 + 18 + 27 + ... + 9n =

Question

What will be an ideal response?

Answer 1

First, we show the statement is true when n = 1.
For n = 1, we get 9 =  = 9.
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.


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