Using the principle of mathematical induction, prove each of the following for all n ϵ N:
To Prove:
Steps to prove by mathematical induction:
Let P(n) be a statement involving the natural number n such that
(i) P(1) is true
(ii) P(k + 1) is true, whenever P(k) is true
Then P(n) is true for all n ϵ N
Therefore,
Let P(n):
Step 1:
P(1) =
Therefore, P(1) is true
Step 2:
Let P(k) is true Then,
P(k):
Now,
=
=
=
=
= P(k + 1)
Hence, P(k + 1) is true whenever P(k) is true
Hence, by the principle of mathematical induction, we have
for all n ϵ N