Suppose that $P(n)$ is a propositional function. Determine for which positive integers $n$ the statement $P(n)$ must be true, and justify your answer, if
a) $P(1)$ is true; for all positive integers $n$, if $P(n)$ is true, then $P(n + 2)$ is true.
b) $P(1)$ and $P(2)$ are true; for all positive integers $n$, if $P(n)$ and $P(n + 1)$ are true, then $P(n + 2)$ is true.
c) $P(1)$ is true; for all positive integers $n$, if $P(n)$ is true, then $P(2n)$ is true.
d) $P(1)$ is true; for all positive integers $n$, if $P(n)$ is true, then $P(n + 1)$ is true.