Use mathematical induction to prove the statement is true for all positive integers n.(45)n = 45n

Question

What will be an ideal response?

Answer 1

Answers may vary. Possible answer:
First we show that the statement is true when n = 1.
 For n = 1, we get (45)1  = 45?1
 Since 45?1 = (45)1  , 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: (45)k+1 = 45(k+1)
So we assume that  is true and multiply both sides of the equation by 45.  
(45)k(45) = 45k(45)
(45)k+1 = 45k+5
(45)k+1 = 45(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 (45)n = 45n  is true for all natural numbers n.