DISCRETE MATH: LECTURE 11
DR. DANIEL FREEMAN
1. Chapter 5.2 Mathematical Induction I
Principle of Mathematical Induction
Let P (n) be a property that is defined for all integers n. Suppose the following two
statements are true.
1. P (a) is true.
2. For all integers k ≥ a, if P (k) is true then P (k + 1) is true. n.
Then for all integers n ≥ a, P (n) is true.
This is illustrated by the following chain of implications:
P (a) is true ⇒ P (a + 1) is true ⇒ P (a + 2) is true ⇒ P (a + 3) is true ⇒ ...
Proving a statement by mathematical induction is a two step process the first step is
called the basis step, the second step is called the inductive step.
Principle of Mathematical Induction
Consider a statement of the form, ”For all integers n ≥ a, a property P (n) is true.” To
prove such a statement, perform the following two steps:
1. (basis step) Prove that P (a) is true.
2. (inductive step) Let k ≥ a and assume P (k) is true. Prove that P (k + 1) is true.
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DR. DANIEL FREEMAN
Theorem 1.1. The sum of the first n positive integers is given by:
n
X
n(n + 1)
i=
2
i=1
Theorem 1.2. For any real number r except 1, and any integer n ≥ 0, the sum of the first
n elements of a geometric series is given by:
n
X
rn+1 − 1
ri =
r−1
i=1
DISCRETE MATH: LECTURE 11
In class work.
(1) Prove for all integers n ≥ 2 that
n−1
X
i(i + 1) =
i=1
n(n − 1)(n + 1)
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(2) Prove for all integers n ≥ 1 that
n
X
i(i!) = (n + 1)! − 1
i=1
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DR. DANIEL FREEMAN
In class work.
(1) Prove for all integers n ≥ 0 that
n+1
X
i · 2i = n · 2n+2 + 2
i=1
(2) Prove for all integers n ≥ 1 that
n
X
i=1
i
1
=
i(i + 1)
i+1
DISCRETE MATH: LECTURE 11
2. Chapter 5.3 Mathematical Induction II
Theorem 2.1. For all integers n ≥ 0, 22n − 1 is divisible by 3.
Theorem 2.2. For all integers n ≥ 3, 2n + 1 < 2n .
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DR. DANIEL FREEMAN
In class work.
(1) Prove for all integers n ≥ 0 that,
5n − 1 is divisible by 4
(2) Prove for all integers n ≥ 2 that
2n < (n + 1)!
DISCRETE MATH: LECTURE 11
In class work.
(1) Prove for all integers n ≥ 0 that,
7n − 1 is divisible by 6
(2) Prove for all integers n ≥ 5 that
n2 < 2n
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