This preview shows pages 1–3. Sign up to view the full content.
MATH 135
Fall 2008
Lectures IV/V Notes
Mathematical Induction
The second technique of proof at which we’ll look is
mathematical induction
. This is a technique that
is normally used to prove that a statement is true for all positive integers
n
.
Motivation
Why would we want to do this? Suppose that we had a sequence
{
x
n
}
deﬁned by
x
1
= 1 and
x
r
+1
= 2
x
r
+ 1 for
r
≥
1.
What is
x
2
?
x
3
?
x
4
?
Do you see a pattern? (Beware of patterns!)
What if we wanted to know
x
2008
? What would we have to do?
It doesn’t make sense to calculate
x
2
through to
x
2007
individually, and it does look like we have
a pattern. Is there a way to prove that
x
n
= 2
n

1 for all positive integers
n
? This would allow us
to be able to state the value of
x
2008
or
x
23487
or whatever.
This is where induction comes in.
Technical Details
Induction is a technique for proving statements of the form
P
(
n
) where
n
∈
P
.
(For example,
P
(
n
) = “
x
n
= 2
n

1”.)
This technique relies on
Principle of Mathematical Induction (POMI)
Let
P
(
n
) be a statement that depends on
n
∈
P
.
If
(i)
P
(1) is true, and
(ii)
P
(
k
) is true
⇒
P
(
k
+ 1) is true,
then
P
(
n
) is true for all
n
∈
P
.
Why does this work? Suppose that we know (i) and (ii) to be true about
P
(
n
).
(i) says
P
(1) is true
(ii) says “If
P
(1) is true, then
P
(2) is true” so
P
(2) is true
(ii) then says “If
P
(2) is true, then
P
(3) is true” so
P
(3) is true
Following this along,
P
(
n
) is true for all
n
∈
P
A couple of analogies: dominos, robot climbing ladder
Proofs by Induction
There are three parts to a proof by induction:
i)
Base Case
– Prove statement for smallest admissible value of
n
(usually
n
= 1)
ii)
Induction Hypothesis
– Suppose that
P
(
k
) is true for some
k
∈
P
iii)
Induction Conclusion
– Prove that
P
(
k
+ 1) is true based on the hypothesis that
P
(
k
) is true
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentExample
A sequence
{
x
n
}
is deﬁned by
x
1
= 1 and
x
r
+1
= 2
x
r
+ 1 for each positive integer
r
≥
1. Prove that
x
n
= 2
n

1 for every positive integer
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 ANDREWCHILDS
 Algebra, Integers, Mathematical Induction

Click to edit the document details