Section 6.2 The Extended Principle of Mathematical Induction
A little exploration shows that the following proposition appears to be true.Proposition.
For each integer
The Extended Principle of Mathematical Induction.
Let
andFor every
with if then
then
Prove that
That is, prove that is true.Prove that for every
with if then That is, prove that if is true, then is true.
Using the Extended Principle of Mathematical Induction.
Let
- Basis step
Prove
- Inductive step
Prove that for every
with if is true, then is true.
We can then conclude that
Proposition 6.2.
For each integer
Proof.
We will use a proof by mathematical induction. For this proof, we let
We first prove thatbe “ ”
For the inductive step, we prove that for all
The goal is to prove that
Now,
Inequalities (9) and (10) show that
and this proves that if