Question

Use the principle of mathematical induction to prove that 2n>n for all positive integers n involving absolute values.

a. True
b. False

Answer 1

True. Using the principle of mathematical induction, we can prove that the statement '2n > n' is True for all positive integers n, which is achieved by confirming it for the base case and proving it holds for every subsequent number using the inductive step.

Step-by-step explanation:

The principle of mathematical induction can indeed be used to prove that 2n > n for all positive integers, including those involving absolute values.

Steps to Prove 2n > n:

1. Base Case: For n=1, 2(1) > 1, which is true.
2. Inductive Step: Assume the statement is true for a positive integer k; that is, assume 2k > k.
3. Consider the next integer k+1. By our assumption, 2k > k. To prove the statement for k+1, consider 2(k+1) = 2k + 2. Since 2k > k and 2 > 1, adding these inequalities gives 2k + 2 > k + 1.
4. Therefore, the statement 2n > n is also true for k+1, completing the inductive step. By the principle of mathematical induction, the statement is true for all positive integers n.

Therefore, the answer to the question is True.