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,…
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