Transcript 5.2 Mathematical Induction I

Discrete Structures
Chapter 5: Sequences, Mathematical Induction, and
Recursion
5.2 Mathematical Induction I
[Mathematical induction is] the standard proof technique in
computer science.
– Anthony Ralston
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Introduction
• Mathematical Induction is one of the more
recently developed techniques of proof in the
history of mathematics.
• It is used to check conjectures about the
outcomes of processes that occur repeatedly an
according to definite patterns.
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Note
• Please make sure that you read through the
proofs and examples in the text book.
• We will be doing different problems in class so
that you will have more examples for
reference. The more you practice, the easier
induction becomes.
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Method of Proof by Mathematical
Induction
• Consider a statement of the form
For all integers n  a, a property P(n) is true.
• Step 1 (basic Step): Show that P(a) is true.
• Step 2 (inductive Step):
– Assume that P(k) is true for all integers k  a.
(inductive hypothesis)
– Show that P(k+1) is true
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Proposition
• Proposition 5.2.1
For all integers n  8, n¢ can obtained using 3¢ and 5¢ coins.
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Theorems
• Theorem 5.2.2 Sum of the First n Integers
For all integers n  1,
1  2  3  ...  n 
n  n  1
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• Theorem 5.2.3 Sum of Geometric Sequence
For any real number r except 1, and any integer n  0,
n
r
i0
i

r
n 1
1
r 1
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Definition
• Closed Form
If a sum with a variable number of terms is shown to
be equal to a formula that does not contain either an
ellipsis or a summation symbol, we say that it is
written in closed form.
1  2  3  ...  n 
n  n  1
2
Closed Form
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Example – pg. 257 #7
• Prove each statement using mathematical
induction. Do not derive them from Theorems
5.2.2 or 5.2.3.
For all integers n  1,
1  6  11  16  ...   5 n  4  
5.2 Mathematical Induction I
n  5n  3
.
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Examples – pg. 257
• Prove each statement by mathematical
induction.
 n  n  1 
11. 1  2 +...  n = 
 , for all integers n  1.
2


2
3
3
3
n 1
14.
 i2
i
 n2
n2
 2, for all integers n  0.
i 1
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Examples – pg. 257
• Use the formula for the sum of the first n integers
and/or the formula for the sum of a geometric
sequence to evaluate the sums or to write them in
closed form.
2 1 . 5  1 0  1 5  2 0  ...  3 0 0
2 6 . 3  3 + 3  ...  3 , w h ere n is an in teg er w ith n  1 .
2
3
n
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