First let me note that it's very good that you made your induction statement (let's call it ) consist of three parts:
is ( is odd) and ( is odd) and ( is even)
The original assertion to prove (let's call it ) is just:
is ( is even)
However, this is not enough to prove the induction step, which is the following statement:
for all , implies
Knowing just is not enough to be able to prove . Therefore, we strengthen the induction statement from to . Now, in proving that implies , we have more to prove (three substatements of instead of just one of ), but we also have a stronger assumption . This technique -- strengthening the induction statement in order for the induction step to go through -- is very common in mathematics.
End of a long digression. Concerning your proof, why do you say "Suppose it is true for k+1"? Inductive step is usually "suppose it is true for k; we need to prove it for k+1", i.e., one has to prove implies for all . So write carefully what you know, i.e., , and what you need to prove, i.e., , and the proof should be pretty straightforward.