This preview shows pages 1–2. Sign up to view the full content.
Induction
Imagine you have a staircase. We need two things in order to be able to walk
up it, no matter how long it is. The ﬁrst step must be safe. Also every step
after a safe step must be safe.
Mathematically we need two things for a proof. We need to establish
that the ﬁrst (step 1) step is true. Also, assuming a step is true (step k), the
next step must be true(step k+1).
We call these the
base case
and the
inductive step
. Let us do this
example:
1 + 2 +
...
+
n
=
n
(
n
+ 1)
2
For the base case, we need to verify that the statement is true for
n
= 1.
LHS
= 1
RHS
= (1
*
2)
/
2 = 1
For the inductive step, we assume that what we want is true for
n
=
k
.
This means:
1 + 2 +
...
+
k
=
k
(
k
+ 1)
2
Now we need to show the statement for
n
=
k
+ 1:
1 + 2 +
...
+
k
+ (
k
+ 1) =
(
k
+ 1)((
k
+ 1) + 1)
2
=
(
k
+ 2)(
k
+ 1)
2
Basically, we start with the left side and use our assumption to get the
right side.
LHS
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '10
 Unknown
 Math, Linear Algebra, Algebra

Click to edit the document details