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.