Use mathematical induction to prove the statement is true for all positive integers n.2n > 2n-1

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, 21 > 21-1.
Since 21-1 = 20 = 1 and 21 > 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: 2k > 2k-1  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: 2k+1 > 2k
So we assume that  is true and multiply both sides of the equation by 2
2k? 2 > 2k-1? 2 
2k+1 > 2(k-1)+1
2k+1 > 2k
So P_k+1 is true if P_k is assumed true. Therefore, by the principle of mathematical induction,  for all natural numbers n.