MAT 3749
Introduction to Analysis
Section 0.2
Mathematical Induction
http://myhome.spu.edu/lauw
Preview of Review

Review Mathematical Induction
References
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Trench, Section 1.2
Howland, Section 1.2
Bloch, Section 2.5
Be sure to pay attention to the
expectations in the class notes.
The Needs…

Theorems involve infinitely many, yet
countable, number of statements.
1 2 
n(n  1)
n
; n N
2
Principle of Mathematical
Induction (PMI)
P(1) P(2) P(3)
P(66) P(67) P(68)
PMI: It suffices to show
1. P(1) is true.
2. If P(k) is true, then P(k+1) is also
true, for all k.
Principle of Mathematical
Induction (PMI)
P(1) P(2) P(3)
P(66) P(67) P(68)
PMI: It suffices to show
1. P(1) is true. (Basic Step)
2. If P(k) is true, then P(k+1) is also
true, for all k (Inductive Step)
Solutions
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In this course, it is extremely important
for you to follow the exact solution format
of using mathematical induction.
Do not skip steps.
You do not have to follow this format in
other courses.
Pedagogical Reasons
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In this course, it is extremely important
for you to follow the exact solution format
of using mathematical induction.
Do not skip steps.
Pedagogical Reasons

Build-in (extra steps) to help you to
• focus on what you need to prove
• avoid making mistakes
Example 1
Use mathematical induction to prove that
1 3 5 
2
2
2
(n  1)(2n  1)(2n  3)
 (2n  1) 
3
2
whenever n is a nonnegative integer.
Checklist A
0,
12  32  52 
 (2n  1) 2 
(n  1)(2n  1)(2n  3)
3
Checklist B
12  32  52 
 (2n  1) 2 
(n  1)(2n  1)(2n  3)
3
Checklist C
12  32  52 
 (2n  1) 2 
(n  1)(2n  1)(2n  3)
3
Checklist C
12  32  52 
 (2n  1) 2 
(n  1)(2n  1)(2n  3)
3
Checklist D
Proof by Mathematical Induction
Declare P(n) and the domain of n.
• Basic Step
• Write down the statement of the first case.
• Do not do any simplifications or algebra on the statement of
the first case.
• Explain why it is true.
• For A=B type, simplify and/ or manipulate each side and
see that they are the same.
• Inductive Step
• Write down the k-th case. This is the inductive hypothesis.
• Write down the (k+1)-th case. This is what you need to prove
to be true.
• For A=B type, we usually start form one side of the equation
and show that it equals to the other side.
• In the process, you need to use the inductive hypothesis.
• Conclude that p(k+1) is true.
• Make the formal conclusion by quoting the PMI
Example 2
Show that for n  1, 2,3,...
n !  2n1
Example 3
Show that for real number r  1,
n 1
 ar
i 0
i


a 1 rn
1 r
,
for n  N