Use mathematical induction to prove the statement is true for all positive integers n.(49)n = 49n
What will be an ideal response?
Answers may vary. Possible answer:
First we show that the statement is true when n = 1.
For n = 1, we get (49)1 = 49?1
Since 49?1 = (49)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,
Pk: 
is assumed true.
On the basis of the assumption that Pk is true, we need to show that Pk+1 is true.
Pk+1: (49)k+1 = 49(k+1)
So we assume that 
is true and multiply both sides of the equation by 49.
(49)k(49) = 49k(49)
(49)k+1 = 49k+9
(49)k+1 = 49(k+1).
The last equation says that Pk+1 is true if Pk is assumed to be true. Therefore, by the principle of mathematical induction, the statement (49)n = 49n is true for all natural numbers n.
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