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Page 1: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Defining and Using Sequences and Series

Section 8.1 beginning on page 410

Page 2: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

SequencesA sequence is an ordered list of numbers. A finite sequence has an end and its domain is the set . The values in the range are called the terms of the sequence.

An infinite sequence is a function that continues without stopping and whose domain is the set of positive integers.

Finite sequence: 2,4,6,8 Infinite sequence: 2,4,6,8,…

A sequence can be specified by an equation, or a rule. For example, both sequences above can be described by the rule π‘Žπ‘›=2𝑛 or 𝑓 (𝑛)=2𝑛

Page 3: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Writing the Terms of a SequenceNote: The domain of a sequence could begin with 0 instead of 1. Unless otherwise indicated, assume the domain of a sequence begins with 1.

Example 1: Write the first six terms of (a) and (b).

a) b)

π‘Ž1=ΒΏ ΒΏ7 First Term

π‘Ž2=ΒΏ ΒΏ9 Second Term

π‘Ž3=ΒΏ ΒΏ11 Third Term

π‘Ž4=ΒΏ ΒΏ13 Fourth Term

π‘Ž5=ΒΏ ΒΏ15 Fifth Term

π‘Ž6=ΒΏ ΒΏ17 Sixth Term

𝑓 (1)=ΒΏ ΒΏ1𝑓 (2)=ΒΏ ΒΏβˆ’3𝑓 (3)=ΒΏ ΒΏ9

𝑓 (4)=ΒΏ ΒΏβˆ’27𝑓 (5)=ΒΏ ΒΏ81𝑓 (6 )=ΒΏ ΒΏβˆ’243

Page 4: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Writing Rules for SequencesExample 2a: Describe the pattern, write the next term, and write a rule for the nth term of the sequence.

,…𝑛=1

𝑛=2 𝑛=3 𝑛=4

How are these terms related to each other? Can they be defined in a similar way?

The terms are all perfect cubes.

The next term would be

To write a rule, relate the terms to their relative position (the n values).

(βˆ’1)3 ,(βˆ’2)3 , (βˆ’3 )3 ,(βˆ’4)3

π‘Žπ‘›=ΒΏ(βˆ’π‘›)3

Page 5: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Writing Rules for SequencesExample 2b: Describe the pattern, write the next term, and write a rule for the nth term of the sequence.

,…𝑛=1

𝑛=2 𝑛=3 𝑛=4

How are these terms related to each other? Can they be defined in a similar way?

Can the terms can be re-written using their relative positions (n values) ?

The next term would be

To write a rule, relate the terms to their relative position (n values).

0 (1),1(2) ,2(3) ,3(4)

π‘Žπ‘›=ΒΏ(π‘›βˆ’1)(𝑛)

Page 6: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Solving a Real-Life Problem** Do not copy, just pay attention.

Page 7: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Writing Rules for SeriesWhen the terms of a sequence are added together, the resulting expression is a series. A series can be finite or infinite.

Finite Series: 2+4+6+8

Infinite Series: 2+4+6+8+…

Summation Notation/Sigma Notation:βˆ‘π‘–=1

4

2𝑖

Summation Notation/Sigma Notation: βˆ‘π‘–=1

∞

2𝑖

is the index (like the relative positon/n-value) and the lower limit of the summation.4 is the upper limit in the finite series, and is the upper limit in the infinite series.

Page 8: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Writing Series Using Summation Notation

Example 4: Write each series using summation notation.

a) b)

** try to re-write the terms using their index (relative position)

25 (1 )+25 (2 )+25 (3 )+β‹― 25 (10)

βˆ‘π‘–=1

10

25 𝑖Ending point

Starting point

11+1

+22+1

+33+1

+44+1

+β‹―

βˆ‘π‘–=1

∞ 𝑖𝑖+1

Ending point

Starting point

Page 9: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Finding the Sum of a SeriesNote: The index of a summation does not have to be , it could be any variable, and it does not have to start at 1.

Example 5: Find the sum βˆ‘π‘˜=4

8

(3+π‘˜2)

ΒΏ (3+ (4 )2)+(3+ (5 )2)+(3+ (6 )2)+(3+ (7 )2)+(3+ (8 )2)

ΒΏ19+28+39+52+67

ΒΏ205For series with many terms, finding the sum by adding the terms can be time consuming. There are formulas for special types of series.

Page 10: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Formulas For Special Series

Sum of n terms of 1:

Sum of first n positive integers:

Sum of squares of first n positive integers:

βˆ‘π‘–

𝑛

1=𝑛

βˆ‘π‘–

𝑛

𝑖=𝑛(𝑛+1)2

βˆ‘π‘–

𝑛

𝑖2=𝑛(𝑛+1)(2𝑛+1)

6

Page 11: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Using a Formula For a Sum

βˆ‘π‘–

𝑛

1=𝑛 βˆ‘π‘–

𝑛

𝑖=𝑛(𝑛+1)2 βˆ‘

𝑖

𝑛

𝑖2=𝑛(𝑛+1)(2𝑛+1)

6

Example 6: How many apples are in the stack in example 3?

The series in example 3 is given by the formula and

βˆ‘π‘–=1

7

𝑖2 ¿𝑛(𝑛+1)(2𝑛+1)

6ΒΏ7(7+1)(2(7 )+1)

6ΒΏ7(8)(15)

6 ΒΏ140

There are 140 apples in the stack.

Page 12: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Monitoring ProgressWrite the first six terms of the sequence.

1) 2) 3)

Find the pattern, write the next term, and write a rule for the nth term of the sequence.

4)

5)

6)

7)

5,6,7,8,9,10 1 ,βˆ’2,4 ,βˆ’8,16 ,βˆ’32 12,23,34,45,56,67

11 ;π‘Žπ‘›=2𝑛+1

35 ;π‘Žπ‘›=𝑛(𝑛+2)

16 ;π‘Žπ‘›=(βˆ’2)π‘›βˆ’1

26 ;π‘Žπ‘›=𝑛2+1

2+1 ,4+1 ,6+1 ,8+1 ,…1(3) ,2 (4 ) ,3 (5 ) ,4 (6 )…

(βˆ’2)0 ,(βˆ’2)1 ,(βˆ’2)2 ,(βˆ’2)3

1+1 ,4+1 ,9+1 ,16+1 ,…

Page 13: Defining and Using Sequences and Series Section 8.1 beginning on page 410.

Monitoring ProgressWrite the series using summation notation.

9) 10)

11) 12)

Find the sum.

13) 14) 15) 16)

βˆ‘π‘–=1

5

8 𝑖 βˆ‘π‘˜=3

7

(π‘˜2βˆ’1) βˆ‘π‘–=1

34

1 βˆ‘π‘˜=1

6

π‘˜

βˆ‘π‘–=1

20

5 𝑖5(1)+5 (2)+5(3)+…+5 (20)

11+1

+44+1

+99+1

+1616+1

+β‹―

βˆ‘π‘–=1

∞ 𝑖2

𝑖2+1

61+62+63+64… βˆ‘π‘–=1

∞

6𝑖 (1+4 )+ (2+4 )+(3+4)…+(8+4)

βˆ‘π‘–=1

8

(𝑖+4)

ΒΏ120 ΒΏ130 ΒΏ34 ΒΏ21