April 30th copyright2009merrydavidson
Happy Birthday to:
4/25 Lauren Cooper
9.1 Sequences & Series
SEQUENCE: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 domain of a sequence is the set of positive integers.
The nth term is used to GENERALIZE about other terms.
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.
“Series” uses + signs.
Arithmetic Sequence Arithmetic Series
3, 8, 13, 18, 23 3 + 8 + 13 + 18 + 23
Given a “rule” for a sequence,
find the 1st 5 terms.
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:
Example 2:
1
2
n
nf
1 1 1 1, , ,
2 4 8 16
Write the first 4 terms of the sequence.
Example 3. Write the first six terms of the sequence if
1 k+11, a 2ka a
1,3,5,7,9,11
Factorial Notation
n! = 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
Factorial Notation
504
n! = n(n – 1)(n – 2)…1
Special case: 0! = 1
9!
5!3!9 8 7 6 5!
5!3!
Summation Notation
1
n
k
rule
The Greek letter sigma, instructs you to add up the terms of the sequence.
52
1
( 1)i
i
Example of sigma notation
Example 4.
52
1
( 1)i
i
Starting with an i value of 1 and ending with an i value of 5, write the series, then add.
2 2 2 2 2(1 1) (2 1) (3 1) (4 1) (5 1)
2 + 5 + 10 + 17 + 26 = 60
Find the sum of:
Example 5.
Starting with an k value of 3 and ending with a k value of 6, write the expanded sum.
62
3
( 1)k
k
2 2 2 2(3 1) (4 1) (5 1) (6 1) Notice: k=3 to k =6 is 4 terms
10 + 17 + 26 + 37 = 90
Find the sum of:
Example 6.
-1 + 0 + 1 + 8 + 27 = 35
43
0
( 1)j
j
Find the sum of:
Notice there are 5 terms here because you are starting at zero.
• A sequence uses comma’s
• A series uses + signs
• Summation notation uses sigma sign
1, 5, 9, 13, 17, …
The common difference is 4
When the difference between successive terms of a sequence is always the same number, the sequence is called arithmetic. In other words, the terms increase (or decrease) by adding a fixed quantity “d”.
Is this sequence arithmetic?
Example 7:
2, -4, 8, -16, 32…
No because we are multiplying by -2 each time.
Is this sequence arithmetic?
Example 8:
-5, 7, 19, 31,…
yes because we are adding 12 each time.
Is the sequence defined by
Sn= 3n + 5 arithmetic?
Example 9:
Let n = 1, n = 2, n = 3, etc to generate the sequence.
8, 11, 14, 17…
yes because we
are adding 3 each time.
Notice that the common
difference is the “slope” of the function.
Therefore linear functions are arithmetic!
Is the sequence defined by
Sn= 4 - n arithmetic?
Example 10:
Let n = 1, n = 2, n = 3, etc to generate the sequence.
3, 2, 1, 0, …
yes because we
are subtracting 1 each time.
d = -1
a1 = 3Therefore linear functions are arithmetic!
Formula for the nth term of Arithmetic Sequence:
“a” is the first term and “d” is the common difference
1 ( 1)na a n d
11) Write the nth term of the sequence 2, 7, 12, 17,…..
Step 1: find the common difference 5
Step 2: write down the formula
1 ( 1)na a n d Step 3: fill in the formula with what you know
2 ( 1)5na n
2 5 5na n
5 3na n
12) Write the nth term of the sequence -12, -9, -6, …..
Step 1: find the common difference 3
Step 2: write down the formula
1 ( 1)na a n d Step 3: fill in the formula with what you know
12 ( 1)3na n
12 3 3na n
3 15na n
Use when you know the first term, number of terms and common difference
Use when you know first term, last term, and number of terms
1
1[2 ( 1) ]
2nS n a n d 1
1( )
2n nS n a a
Notice: Both formulas need first term and number of terms.
SUMMATION FORMULAS:
13) Find the sum of the first 12 terms of:
• Find “d”.
.
4 8 12 16 20
d = 4• Pick which formula
you want to use.
1
1[2 ( 1) ]
2nS n a n d
• Plug and Chug. 12
112[2 4 (12 1) 4 ]
2S
12 6[8 (11) 4 ]S
12 312S
14) Find the indicated partial sum of:• Find the 1st term.
.
a1 = -1
• Pick which formula you want to use. • Plug and Chug.
25
0
(2 1)j
j
• Find the 2nd term.
a2 = 1• Find “d”. d = 2
• Find the last term.
a26 = 51
1
1( )
2n nS n a a
1(26)( 1 51)
2nS
650nS