Use mathematical induction to prove the statement is true for all positive integers n.7n > n
What will be an ideal response?
Answers may vary. Possible answer:
First, we show the statement is true when n = 1.
For n = 1, 71 > 1
So 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: 7k > k is assumed true.
On the basis of the assumption that Pk is true, we need to show that Pk+1 is true.
Pk+1: 7k+1 > k + 1
So we assume that is true and multiply both sides of the equation by 7
7k? 7 > k7
7k+1 > k + 6k
7k+1 > k + 1 since 6k > 1
So Pk+1 is true if Pk is assumed true. Therefore, by the principle of mathematical induction, for all natural numbers n.
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