Post on 17-Jan-2016
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,โฆ
A sequence can be specified by an equation, or a rule. For example, both sequences above can be described by the rule ๐๐=2๐ or ๐ (๐)=2๐
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)
๐1=ยฟ ยฟ7 First Term
๐2=ยฟ ยฟ9 Second Term
๐3=ยฟ ยฟ11 Third Term
๐4=ยฟ ยฟ13 Fourth Term
๐5=ยฟ ยฟ15 Fifth Term
๐6=ยฟ ยฟ17 Sixth Term
๐ (1)=ยฟ ยฟ1๐ (2)=ยฟ ยฟโ3๐ (3)=ยฟ ยฟ9
๐ (4)=ยฟ ยฟโ27๐ (5)=ยฟ ยฟ81๐ (6 )=ยฟ ยฟโ243
Writing Rules for SequencesExample 2a: Describe the pattern, write the next term, and write a rule for the nth term of the sequence.
,โฆ๐=1
๐=2 ๐=3 ๐=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).
(โ1)3 ,(โ2)3 , (โ3 )3 ,(โ4)3
๐๐=ยฟ(โ๐)3
Writing Rules for SequencesExample 2b: Describe the pattern, write the next term, and write a rule for the nth term of the sequence.
,โฆ๐=1
๐=2 ๐=3 ๐=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)
๐๐=ยฟ(๐โ1)(๐)
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+โฆ
Summation Notation/Sigma Notation:โ๐=1
4
2๐
Summation Notation/Sigma Notation: โ๐=1
โ
2๐
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 )+โฏ 25 (10)
โ๐=1
10
25 ๐Ending point
Starting point
11+1
+22+1
+33+1
+44+1
+โฏ
โ๐=1
โ ๐๐+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 โ๐=4
8
(3+๐2)
ยฟ (3+ (4 )2)+(3+ (5 )2)+(3+ (6 )2)+(3+ (7 )2)+(3+ (8 )2)
ยฟ19+28+39+52+67
ยฟ205For 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:
โ๐
๐
1=๐
โ๐
๐
๐=๐(๐+1)2
โ๐
๐
๐2=๐(๐+1)(2๐+1)
6
Using a Formula For a Sum
โ๐
๐
1=๐ โ๐
๐
๐=๐(๐+1)2 โ
๐
๐
๐2=๐(๐+1)(2๐+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
โ๐=1
7
๐2 ยฟ๐(๐+1)(2๐+1)
6ยฟ7(7+1)(2(7 )+1)
6ยฟ7(8)(15)
6 ยฟ140
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 ,โ2,4 ,โ8,16 ,โ32 12,23,34,45,56,67
11 ;๐๐=2๐+1
35 ;๐๐=๐(๐+2)
16 ;๐๐=(โ2)๐โ1
26 ;๐๐=๐2+1
2+1 ,4+1 ,6+1 ,8+1 ,โฆ1(3) ,2 (4 ) ,3 (5 ) ,4 (6 )โฆ
(โ2)0 ,(โ2)1 ,(โ2)2 ,(โ2)3
1+1 ,4+1 ,9+1 ,16+1 ,โฆ
Monitoring ProgressWrite the series using summation notation.
9) 10)
11) 12)
Find the sum.
13) 14) 15) 16)
โ๐=1
5
8 ๐ โ๐=3
7
(๐2โ1) โ๐=1
34
1 โ๐=1
6
๐
โ๐=1
20
5 ๐5(1)+5 (2)+5(3)+โฆ+5 (20)
11+1
+44+1
+99+1
+1616+1
+โฏ
โ๐=1
โ ๐2
๐2+1
61+62+63+64โฆ โ๐=1
โ
6๐ (1+4 )+ (2+4 )+(3+4)โฆ+(8+4)
โ๐=1
8
(๐+4)
ยฟ120 ยฟ130 ยฟ34 ยฟ21