Prove or disprove the statement by Mathematical Induction

Question

For ∀ n∈Z^+ , 2^n + 1 is divisible by 3

step by step solutions would be helpful

Answer 1

For n = 2

2² + 1

4 + 1

5

Answer 2

Show that the base case (n = 1) works:

When n = 1, 2ⁿ + 1 = 3 which is a multiple of 3.

So that checks out.

Now let's assume that P(k) works and show that whether P(k+1) works or not

So if P(k) then 2^k + 1 is a multiple of 3, then 2 * (2^k + 1) is a multiple of 3.

Therefore, after distributing the 2 we get 2^k+1 + 2 is a multiple of 3.

However P(k+1) is the statement 2^k+1 + 1 is a multiple of 3.

2^k+1 + 2 is a multiple of 3 implies that 2^k+1 + 1 can't be a multiple of 3

Therefore, the statement is disproved