Wrong Solution?

Question

So I got this solution from the web cause my textbook doesn't come with steps, is the second summation wrong though? But the final answer is correct.

For $n=0$, $P(X=2^0+1) = P(X=2)$

$2\ne odd$

Answer 1

Read it carefully! The book doesn't state $X = 2^n + 1$ but $X=2n+1$. Otherwise the third equality sign is false and you wouldn't get e.g. $X=7$ because there is no $n\in \Bbb N$ s.t. $2^n + 1 = 7$