Answer
--Determining:
-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.
Work Step by Step
a) The inductive step here allows us to conclude that P(3),P(5), . . . are all true,
-- but we can conclude nothing about P(2), P(4), . . . .
b) P(n) is true for all positive integers n, using
strong induction.
c) The inductive step here enables us to
conclude that P(2), P(4), P(8), P(16), …are all true,
but we can conclude nothing about P(n) when n is not a power of 2.
d) This is mathematical induction; we can conclude that P(n)
is true for all positive integers n.
