For what positive integer values $n$, is $2^n+1$ divisible by 3

Question

For what positive integer values $n$, is $2^n+1$ divisible by 3, but I am not sure how to proceed.

The only thing I can depict is that: $2^n-1 \equiv 0 \pmod{3} \implies 2^n \equiv -1 \pmod{3}$

Where do I go from here?

Answer 1

Hint:

$$2 \equiv -1 \pmod 3$$

$$2^n \equiv (-1)^n \pmod 3$$

Answer 2

Let $a\in\{0,1\}$. Then $$ 1 + 2^{2n+a} = 1 + 2^a4^n = 1 + 2^a(1+3)^n = 1 + 2^a\sum_{k=0}^n\binom n k 3^k = 1 + 2^a + 3\sum_{k=1}^n\binom n k3^{k-1}. $$