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,β¦
A sequence can be specified by an equation, or a rule. For example, both sequences above can be described by the rule ππ=2π or π (π)=2π
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)
π1=ΒΏ ΒΏ7 First Term
π2=ΒΏ ΒΏ9 Second Term
π3=ΒΏ ΒΏ11 Third Term
π4=ΒΏ ΒΏ13 Fourth Term
π5=ΒΏ ΒΏ15 Fifth Term
π6=ΒΏ ΒΏ17 Sixth Term
π (1)=ΒΏ ΒΏ1π (2)=ΒΏ ΒΏβ3π (3)=ΒΏ ΒΏ9
π (4)=ΒΏ ΒΏβ27π (5)=ΒΏ ΒΏ81π (6 )=ΒΏ ΒΏβ243
Writing Rules for SequencesExample 2a: Describe the pattern, write the next term, and write a rule for the nth term of the sequence.
,β¦π=1
π=2 π=3 π=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).
(β1)3 ,(β2)3 , (β3 )3 ,(β4)3
ππ=ΒΏ(βπ)3
Writing Rules for SequencesExample 2b: Describe the pattern, write the next term, and write a rule for the nth term of the sequence.
,β¦π=1
π=2 π=3 π=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)
ππ=ΒΏ(πβ1)(π)
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+β¦
Summation Notation/Sigma Notation:βπ=1
4
2π
Summation Notation/Sigma Notation: βπ=1
β
2π
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 )+β― 25 (10)
βπ=1
10
25 πEnding point
Starting point
11+1
+22+1
+33+1
+44+1
+β―
βπ=1
β ππ+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 βπ=4
8
(3+π2)
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.
Formulas For Special Series
Sum of n terms of 1:
Sum of first n positive integers:
Sum of squares of first n positive integers:
βπ
π
1=π
βπ
π
π=π(π+1)2
βπ
π
π2=π(π+1)(2π+1)
6
Using a Formula For a Sum
βπ
π
1=π βπ
π
π=π(π+1)2 β
π
π
π2=π(π+1)(2π+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
βπ=1
7
π2 ΒΏπ(π+1)(2π+1)
6ΒΏ7(7+1)(2(7 )+1)
6ΒΏ7(8)(15)
6 ΒΏ140
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 ,β2,4 ,β8,16 ,β32 12,23,34,45,56,67
11 ;ππ=2π+1
35 ;ππ=π(π+2)
16 ;ππ=(β2)πβ1
26 ;ππ=π2+1
2+1 ,4+1 ,6+1 ,8+1 ,β¦1(3) ,2 (4 ) ,3 (5 ) ,4 (6 )β¦
(β2)0 ,(β2)1 ,(β2)2 ,(β2)3
1+1 ,4+1 ,9+1 ,16+1 ,β¦
Monitoring ProgressWrite the series using summation notation.
9) 10)
11) 12)
Find the sum.
13) 14) 15) 16)
βπ=1
5
8 π βπ=3
7
(π2β1) βπ=1
34
1 βπ=1
6
π
βπ=1
20
5 π5(1)+5 (2)+5(3)+β¦+5 (20)
11+1
+44+1
+99+1
+1616+1
+β―
βπ=1
β π2
π2+1
61+62+63+64β¦ βπ=1
β
6π (1+4 )+ (2+4 )+(3+4)β¦+(8+4)
βπ=1
8
(π+4)
ΒΏ120 ΒΏ130 ΒΏ34 ΒΏ21
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