Use mathematical induction to prove the statement is true for all positive integers n.9 + 18 + 27 + . . . + 9n =

Question

What will be an ideal response?

Answer 1

Answers may vary. Possible answer:
First, we show the statement is true when n = 1.
For n = 1, we get 9 =  
 Since  =  = 9, P₁ is true and the first condition for the principle of induction is satisfied.
Next, we assume the statement holds for some unspecified natural number k. That is,
P_k:   is assumed true.
On the basis of the assumption that P_k is true, we need to show that P_k+1 is true. 
P_k+1: 
So we assume that  is true and add the next term,  to both sides of the equation.

9 + 18 + 27 + ... + 9k + 9(k + 1) =  + 9(k + 1)
9 + 18 + 27 + ... + 9k + 9(k + 1) = 9

9 + 18 + 27 + ... + 9k + 9(k + 1) = 
The last equation says that P_k+1 is true if P_k is assumed to be true. Therefore, by the principle of mathematical induction, the statement 9 + 18 + 27 + ... + 9n =   is true for all natural numbers n.