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 𝑎𝑛=2𝑛 or 𝑓 (𝑛)=2𝑛

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)

𝑎1=¿ ¿7 First Term

𝑎2=¿ ¿9 Second Term

𝑎3=¿ ¿11 Third Term

𝑎4=¿ ¿13 Fourth Term

𝑎5=¿ ¿15 Fifth Term

𝑎6=¿ ¿17 Sixth Term

𝑓 (1)=¿ ¿1𝑓 (2)=¿ ¿−3𝑓 (3)=¿ ¿9

𝑓 (4)=¿ ¿−27𝑓 (5)=¿ ¿81𝑓 (6 )=¿ ¿−243

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).

(−1)3 ,(−2)3 , (−3 )3 ,(−4)3

𝑎𝑛=¿(−𝑛)3

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)

𝑎𝑛=¿(𝑛−1)(𝑛)

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

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:∑𝑖=1

4

2𝑖

Summation Notation/Sigma Notation: ∑𝑖=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.

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 )+⋯ 25 (10)

∑𝑖=1

10

25 𝑖Ending point

Starting point

11+1

+22+1

+33+1

+44+1

+⋯

∑𝑖=1

∞ 𝑖𝑖+1

Ending point

Starting point

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 ∑𝑘=4

8

(3+𝑘2)

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

¿19+28+39+52+67

¿205For series with many terms, finding the sum by adding the terms can be time consuming. There are formulas for special types of series.

Formulas For Special Series

Sum of n terms of 1:

Sum of first n positive integers:

Sum of squares of first n positive integers:

∑𝑖

𝑛

1=𝑛

∑𝑖

𝑛

𝑖=𝑛(𝑛+1)2

∑𝑖

𝑛

𝑖2=𝑛(𝑛+1)(2𝑛+1)

6

Using a Formula For a Sum

∑𝑖

𝑛

1=𝑛 ∑𝑖

𝑛

𝑖=𝑛(𝑛+1)2 ∑

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

∑𝑖=1

7

𝑖2 ¿𝑛(𝑛+1)(2𝑛+1)

6¿7(7+1)(2(7 )+1)

6¿7(8)(15)

6 ¿140

There are 140 apples in the stack.

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 ,−2,4 ,−8,16 ,−32 12,23,34,45,56,67

11 ;𝑎𝑛=2𝑛+1

35 ;𝑎𝑛=𝑛(𝑛+2)

16 ;𝑎𝑛=(−2)𝑛−1

26 ;𝑎𝑛=𝑛2+1

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

(−2)0 ,(−2)1 ,(−2)2 ,(−2)3

1+1 ,4+1 ,9+1 ,16+1 ,…

Monitoring ProgressWrite the series using summation notation.

9) 10)

11) 12)

Find the sum.

13) 14) 15) 16)

∑𝑖=1

5

8 𝑖 ∑𝑘=3

7

(𝑘2−1) ∑𝑖=1

34

1 ∑𝑘=1

6

𝑘

∑𝑖=1

20

5 𝑖5(1)+5 (2)+5(3)+…+5 (20)

11+1

+44+1

+99+1

+1616+1

+⋯

∑𝑖=1

∞ 𝑖2

𝑖2+1

61+62+63+64… ∑𝑖=1

∞

6𝑖 (1+4 )+ (2+4 )+(3+4)…+(8+4)

∑𝑖=1

8

(𝑖+4)

¿120 ¿130 ¿34 ¿21