(CSC 102)

Lecture 23

Discrete Structures

Previous Lecture Summery

Sequences

Alternating Sequence

Summation Notation

Product Notation

Properties of Sequences

Change of Variable

Factorial Notations

Change Of Variable

Observe thet

And also

This equation illustrates the fact that the symbol used to represent the index of a summation can be replaced by any other symbol as long as the replacement is made in each location where the symbol occurs. As a consequence, the index of a summation is called a dummy variable. A dummy variable is a symbol that derives its entire meaning from its local context. Outside of that context (both before and after), the symbol may have another meaning entirely.

Transforming a Sum by a Change of Variable

Upper Limit Appears in the Expression to Be Summed

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Factorial Notations

Examples

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Special Factorial Cases

Example

Mathematical Induction I

Mathematical Induction

Principle of Mathematical Induction

Method of proof

Finding Terms of Sequences

Sum of Geometric Series

Introduction

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Cont…

Principal Of Mathematical Induction

Method Of Proof

Finding Terms of Sequences

Explicit formula

Propositions

Proposition: For all integers n ≥ 1,1 + 2+· · ·+n = n(n + 1)/2

Proof:

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Sum of Geometric Series

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Deducing Additional Formula

As with the formula for the sum of the first n integers, there is a way to think of the formula for the sum of the terms of a geometric sequence that makes it seem simple and intuitive. Let

Equating the right-hand sides of equations and dividing by r − 1 gives

Cont…

Lecture Summery

Change of Variable

Factorial Notations

Mathematical Induction

Method of proof

Finding Terms of Sequences

Sum of Geometric Series