MATH 42 — Induction
The principle of mathematical induction refers to a sequence of mathematical statements that
we can enumerate with natural numbers. For example, such a sequence of statements is:
•
1 = 1
2
•
1 + 3 = 2
2
•
1 + 3 + 5 = 3
2
•
1 + 3 + 5 + 7 = 4
2
•
and so on...
which can be summarized as the family of statements
1 + 3 + 5 +
· · ·
+ (2
n

1) =
n
2
for every
n .
The principle of mathematical induction is a means for proving such statements. The steps of
an induction proof are as follows.
1. Prove that the first statement is true.
2. Assume the
k
th
statement is true.
3. Prove that the (
k
+ 1)
st
statement is true.
If the family of statements is a ladder that you’re trying to climb all the way to the
n
th
rung, then
Steps 2 and 3 say that if you’re on the
k
th
rung then you can get to the (
k
+ 1)
st
rung. The first
step says that you can begin your climb!
Here is an induction proof that
∑
n
i
=1
(2
i

1) =
n
2
.
Step 1: The statement is clearly true when
n
= 1.
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 Winter '07
 Staff
 Linear Algebra, Algebra, Differential Calculus, Natural Numbers, Mathematical Induction, Natural number, Mathematical logic, Mathematical proof, Induction Proof

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