Use mathematical induction to prove the statement is true for all positive integers n.4n > n

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, 41 > 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,
P_k: 4k > 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: 4k+1 > k + 1
So we assume that  is true and multiply both sides of the equation by 4
4k? 4 > k4
4k+1 > k + 3k
4k+1 > k + 1 since 3k > 1
So P_k+1 is true if P_k is assumed true. Therefore, by the principle of mathematical induction,  for all natural numbers n.