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Arithmetic Sequences and Series
Section 9-2
2
Objectives
• Use sequence notation to find terms of any sequence
• Use summation notation to write sums
• Use factorial notation
• Find sums of infinite series
3
9.1 Sequences & SeriesSEQUENCE: A list that is
ordered so that it has a 1st term, a 2nd term, a 3rd term and so on.
example: 1, 5, 9, 13, 17, …a1 = 1; a2
= 5; a3 = 9, etc.
The nth term is denoted by: an
The nth term is used to GENERALIZE about other terms.
4
The three dots mean that this sequence is INFINITE.
example: 1, 5, 9, 13, 17, …
example: 2, -9, 28, -65, 126
This is a FINITE sequence.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
Given a “rule” for a sequence,
find the sequence.
2
4
14 1 7
2f
211
2nf x
2
1
1 11 1
2 2f 2
2
12 1 1
2f
2
3
1 73 1
2 2f
2
5
1 235 1
2 2f
1 7 23,1, ,7,
2 2 2
EXAMPLE 1:
6
Write the first four terms of the sequence whose nth term is given by:
12
)1(
n
n
na
17
19
15
13
1
4
3
2
1
a
a
a
a
7
Recursion Formula
• Defines the nth term of a sequence as a function of the previous term.
235 11 nn aaanda
51 a
23 12 aa
172)5(32 a
532)17(33 a
1612)53(34 a
8
Find the first four terms
523 11 nn aaanda
123
59
27
11
4
3
2
1
a
a
a
a
9
Factorial Notationn! = n(n – 1)(n – 2)…1
Special case: 0! = 1
8! 8 7 6 5 4 3 2 1 8 Math/prb/4/enter
= 40,320
10
A sequence is arithmetic if the differences between consecutive terms are the same.
4, 9, 14, 19, 24, . . .
9 – 4 = 5
14 – 9 = 5
19 – 14 = 5
24 – 19 = 5
arithmetic sequence
The common difference, d, is 5.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
Example: Find the first five terms of the sequence and determine if it is arithmetic.
an = 1 + (n – 1)4
This is an arithmetic sequence.
d = 4
a1 = 1 + (1 – 1)4 = 1 + 0 = 1
a2 = 1 + (2 – 1)4 = 1 + 4 = 5
a3 = 1 + (3 – 1)4 = 1 + 8 = 9
a4 = 1 + (4 – 1)4 = 1 + 12 = 13
a5 = 1 + (5 – 1)4 = 1 + 16 = 17
12
The nth term of an arithmetic sequence has the form
an = dn + c
where d is the common difference and c = a1 – d.
2, 8, 14, 20, 26, . . . .
d = 8 – 2 = 6
a1 = 2 c = 2 – 6 = – 4
The nth term is: an = 6n – 4.
13
a1 – d =
Example: Find the formula for the nth term of an arithmetic sequence whose common difference is 4 and whose first term is 15. Find the first five terms of the sequence.
an = dn + c
= 4n + 11
15,
d = 4
a1 = 15 19, 23, 27, 31.
The first five terms are
15 – 4 = 11
14
The sum of a finite arithmetic sequence with n terms is given by
1( ).2n nnS a a
5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = ?
( )501 2755 )0 5(552nS
n = 10
a1 = 5 a10 = 50
15
The sum of the first n terms of an infinite sequence is called the nth partial sum.
1( )2n nnS a a
( )190 25(184) 4602
50 6 0nS
a1 = – 6
an = dn + c = 4n – 10
Example: Find the 50th partial sum of the arithmetic sequence – 6, – 2, 2, 6, . . .
d = 4 c = a1 – d = – 10
a50 = 4(50) – 10 = 190
16
• The sum of the first n terms of a sequence is represented by the summation notation
• Where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.
Summation Notation
n
n
ii aaaaaa
...43211
17
Consider the infinite sequence a1, a2, a3, . . ., ai, . . ..
1. The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence.
1
n
ii
a
a1 + a2 + a3 + . . . + an
2. The sum of all the terms of the infinite sequence is called an infinite series.
1i
i
a
a1 + a2 + a3 + . . . + ai + . . .
18
Example
5
1
)25(i
i• Expand and evaluate the sum:
Solution:
5
53113
)25(5
1
i
i
19
100
1
2i
n
Example: Find the partial sum.
2( ) 2( ) 2( ) 2( )1 2 3 100 2 4 6 200
a1 a100
100 1 10010( ) 2( )02 0
2 20nS a a
50(202) 10,100
20
Homework
• WS 13-4