Question

Use mathematical induction to prove that for all natural numbers n, 3^n - 1 is an even number.

a) True
b) False

Answer 1

To prove that for all natural numbers n, 3^n - 1 is an even number, we can use mathematical induction.

Step-by-step explanation:

To prove that for all natural numbers n, 3^n - 1 is an even number using mathematical induction, we'll follow these steps:

1. Base case: Start by checking for n = 1. When n = 1, 3^n - 1 = 3^1 - 1 = 2, which is indeed an even number.
2. Inductive hypothesis: Assume that for some k, 3^k - 1 is an even number.
3. Inductive step: Show that if the statement holds for k, it also holds for k + 1. So, we need to prove that 3^(k + 1) - 1 is an even number using the assumption from the previous step.
4. Conclusion: By the principle of mathematical induction, we can conclude that for all natural numbers n, 3^n - 1 is an even number.

Therefore, the statement is true.