Chapter 5: Sequences, Mathematical Induction and Recursion

Discrete Mathematics and Applications

Yan Zhang (yzhang16@usf.edu)

Department of Computer Science and Engineering, USF

5.1 Sequences

Sequences

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• A sequence is a function.

• Function – Every element in is mapped to only one element in , such that .

Domain: A Codomain: B𝑓 : 𝐴→𝐵

Not allowed

Definition

• A sequence is a function.

• Domain is either all the integers between two given integers or all the integers greater than or equal to a given integer.

Sequences

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Domain

Codomain

Domain

Codomain

Finite sequences:

Infinite sequences:

(read “ sub ”) is called a term. in is called a subscript or index.

Initial term: Final term:

Sequence Example 1Finding Terms of Sequences Given by Explicit Formulas

Define sequences and by the following explicit formulas:

Compute the first five terms of both sequences.

Solution:

,,,

Note: all terms of both sequences are identical.

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Sequence Example 2Finding an Explicit Formulas to Fit Given Initial Terms

Find an explicit formula for a sequence that has the following initial terms:

Solution:

• Denote the general term of the sequence by and suppose the first term is .

,• An explicit formulator that gives the correct first six terms is:

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Summation Notation

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Domain

Codomain

Given a sequence ,

The summation from equals to of -sub-:

: the index of the summation: the lower limit of the summation: the upper limit of the summation

Summation notation Expanded form

Summation Example – Computing Summations

Let and . Compute the following:

a.

b.

c.

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Summation Example – Changing from Summation Notation to Expanded Form

Write the following summation in expanded form:

Solution:

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Summation Example – Changing from Expanded Form to Summation Notation

Express the following using summation notation:

Solution:

The general term of this summation can be expressed as for integers k from 0 to n.

Hence

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Summation Notation - Recursive DefinitionIf is any integer and , then

Recursive definition is useful to rewrite a summation, – by separating off the final term of a summation – by adding a final term to a summation.

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∑𝑘=𝑚

𝑛

𝑎𝑘=∑𝑘=𝑚

𝑛−1

𝑎𝑘+𝑎𝑛

∑𝑘=𝑚

𝑚

𝑎𝑘=𝑎𝑚

∑𝑘=𝑚

𝑚+1

𝑎𝑘=∑𝑘=𝑚

𝑚

𝑎𝑘+𝑎𝑚+1

∑𝑘=𝑚

𝑚+2

𝑎𝑘=∑𝑘=𝑚

𝑚+1

𝑎𝑘+𝑎𝑚+2

⋯

Recursive Definition Example – Separating Off a Final Term and Adding On a Final Term

a. Rewrite by separating off the final term.

b. Write as a single summation.

Solution:

a.

b.

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Summation Notation - A Telescoping Sum

• In certain sums each term is a difference of two quantities.

For example:

• When you write such sums in expanded form, you sometimes see that all the terms cancel except the first and the last.

• Successive cancellation of terms collapses the sum like a telescope.

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A Telescoping Sum Example

Some sums can be transformed into telescoping sums, which then can be rewritten as a simple expression.

For instance, observe that

Use this identity to find a simple expression for

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Product Notation

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Domain

Codomain

Given a sequence ,

The product from equals to of -sub-:

A recursive definition for the product notation is the following:

If is any integer, then

Product Notation Example – Computing Products

Compute the following products:

a.

b.

Solution:

a.

b.15

Properties of Summations and Products

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Properties of Summation & Product Exercise

Let and for all integers . Write each of the following expressions as a single summation or product:

a. b.

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Properties of Summation & Product Exercise

Solution:

a.

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Properties of Summation & Product Exercise

Solution:

b.

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Change of Variable

• The index symbol in a summation or product is internal to summation or product.

• The index symbol can be replaced by any other symbol as long as the replacement is made in each location where the symbol occurs.

and

• As a consequence, the index of a summation or a product is called a dummy variable.

• A dummy variable is a symbol that derives its entire meaning from its local context. Outside of that context, the symbol may have another meaning entirely.

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Change of Variable Exercise 1

summation: change of variable:

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Change of Variable Exercise 1 - Solution

Solution:• First calculate the lower and upper limits of the new

summation:

Thus the new sum goes from j = 1 to j = 7.

• Next calculate the general term of the new summation by replacing each occurrence of k by an expression in j :

• Finally, put the steps together to obtain

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Factorial Notation

A recursive definition for factorial is the following: Given any nonnegative integer ,

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Simplify the following expressions:

a. b. c.

d. e.

Solution:

a.

b.

Factorials Exercise

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d.

e.

Factorials Exercise Solution

c.

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“n Choose r ” Notation

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Chose 2 objects from {a,b,c,d}:

{a, b}, {a, c}, {a, d}, {b, c}, {b,d}, and {c, d}.

“n Choose r” Exercise

Use the formula for computing to evaluate the following expressions:

a. b. c.

Solution:

a.

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b.

c.

“n Choose r” Exercise

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n Choose r Properties

• for

• for • for • for

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ExerciseProve that for all nonnegative integers and with ,

.

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ExerciseProve that for all nonnegative integers and with ,

.

Hint:

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