Use mathematical induction to prove the statement is true for all positive integers n.(69)n = 69n

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 (69)1  = 69?1
 Since 69?1 = (69)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: (69)k+1 = 69(k+1)
So we assume that  is true and multiply both sides of the equation by 69.  
(69)k(69) = 69k(69)
(69)k+1 = 69k+9
(69)k+1 = 69(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 (69)n = 69n  is true for all natural numbers n.