Intro to Series and Sequences
-
Upload
george-brown -
Category
Documents
-
view
223 -
download
0
Transcript of Intro to Series and Sequences
-
7/29/2019 Intro to Series and Sequences
1/15
11.1 An Introduction toSequences & Series
p. 651
-
7/29/2019 Intro to Series and Sequences
2/15
Sequence:
A list of ordered numbers separated bycommas.
Each number in the list is called a term.
For Example:
Sequence 1 Sequence 2
2,4,6,8,10 2,4,6,8,10,
Term 1, 2, 3, 4, 5 Term 1, 2, 3, 4, 5
Domain relative position of each term (1,2,3,4,5)Usually begins with position 1 unless otherwisestated.
Range the actual terms of the sequence(2,4,6,8,10)
-
7/29/2019 Intro to Series and Sequences
3/15
Sequence 1 Sequence 2
2,4,6,8,10 2,4,6,8,10,
A sequence can be finiteor infinite.
The sequence hasa last term or finalterm.
(such as seq. 1)
The sequencecontinues withoutstopping.
(such as seq. 2)
Both sequences have a general rule: an = 2n wheren is the term # and an is the nth term.
The general rule can also be written in function
notation: f(n) = 2n
-
7/29/2019 Intro to Series and Sequences
4/15
Examples:
Write the first 6
terms of an=5-n. a1=5-1=4
a2=5-2=3
a3=5-3=2
a4=5-4=1
a5=5-5=0
a6=5-6=-1
4,3,2,1,0,-1
Write the first 6terms of an=2n.
a1=21=2
a2=22=4
a3=23=8
a4=24=16
a5=25=32
a6=26=64
2,4,8,16,32,64
-
7/29/2019 Intro to Series and Sequences
5/15
Examples: Write a rule for the nth term.
The seq. can bewritten as:
Or, an=2/(5n)
The seq. can be
written as:
2(1)+1, 2(2)+1, 2(3)+1,2(4)+1,
Or, an=2n+1
,...625
2,
125
2,
25
2,
5
2.a
,...5
2,
5
2,
5
2,
5
24321
,...9,7,5,3.b
-
7/29/2019 Intro to Series and Sequences
6/15
Example: write a rule for the nth term.
2,6,12,20,
Can be written as:
1(2), 2(3), 3(4), 4(5),
Or, an=n(n+1)
-
7/29/2019 Intro to Series and Sequences
7/15
Graphing a Sequence
Think of a sequence as ordered pairs forgraphing. (n , an)
For example: 3,6,9,12,15
would be the ordered pairs (1,3), (2,6),(3,9), (4,12), (5,15) graphed like points in a
scatter plot* Sometimes it helps to find the rule first
when you are not given every term in afinite sequence.
Term # Actual term
-
7/29/2019 Intro to Series and Sequences
8/15
Series
The sum of the terms in a sequence.
Can be finite or infinite
For Example:
Finite Seq. Infinite Seq.
2,4,6,8,10 2,4,6,8,10,
Finite Series Infinite Series2+4+6+8+10 2+4+6+8+10+
-
7/29/2019 Intro to Series and Sequences
9/15
Summation Notation Also called sigma notation
(sigma is a Greek letter meaning sum)
The series 2+4+6+8+10 can be written as:
i is called the index of summation
(its just like the n used earlier).Sometimes you will see an n or k here instead of i.
The notation is read:
the sum from i=1 to 5 of 2i
5
12i
i goes from 1
to 5.
-
7/29/2019 Intro to Series and Sequences
10/15
Summation Notation for anInfinite Series
Summation notation for the infinite series:
2+4+6+8+10+ would be written as:
Because the series is infinite, you must use ifrom 1 to infinity () instead of stopping at
the 5th term like before.
1
2i
-
7/29/2019 Intro to Series and Sequences
11/15
Examples: Write each series insummation notation.
a. 4+8+12++100
Notice the series canbe written as:
4(1)+4(2)+4(3)++4(25)
Or 4(i) where i goesfrom 1 to 25.
Notice the seriescan be written as:
25
1
4i
...5
4
4
3
3
2
2
1
. b
...14
413
312
211
1
.to1fromgoeswhere
1
Or,
ii
i
1 1i
i
-
7/29/2019 Intro to Series and Sequences
12/15
Example: Find the sum of theseries.
k goes from 5 to 10.
(52+1)+(62+1)+(72+1)+(82+1)+(92+1)+(102+1)
= 26+37+50+65+82+101
= 361
10
5
2
1k
-
7/29/2019 Intro to Series and Sequences
13/15
Special Formulas (shortcuts!)
n
n
i
1
12
)1(
1
nni
n
i
6
)12)(1(
1
2
nnni
n
i
-
7/29/2019 Intro to Series and Sequences
14/15
Example: Find the sum.
Use the 3rd
shortcut!
10
1
2
i
i
6
)12)(1( nnn
6
)110*2)(110(10
6
21*11*10 385
6
2310
-
7/29/2019 Intro to Series and Sequences
15/15
Assignment
http://localhost/var/www/apps/conversion/tmp/scratch_2/honors_algebra_2_homework_assign.htmhttp://localhost/var/www/apps/conversion/tmp/scratch_2/honors_algebra_2_homework_assign.htm