seq and series notes
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Sequence
A sequence is a set of numbers
in a specific order
Infinite sequence
Finite sequence
a1 , a2 , a3 , a4 ,..., an ,...
a1 , a2 , a3 , a4 ,..., an
Sequences –
sets of numbers
Notation:
an represents the formula for finding terms
n term number
a4 is the notation for the 4th term
a32 is the notation for the 32nd term
Examples:
If an 2n 3, find the first 5 terms.
If an 3n 1, find the 20th term.
.
Ex 1
Find the first four terms of the sequence
an 3n 2
a1 3(1) 2 1
First term
a2 4
Second term
a3 7
Third term
a4 10
Fourth term
Writing Rules for Sequences
We can calculate as many terms as we want as
long as we know the rule or equation for an.
Example:
3, 5, 7, 9, ___ , ___,……. _____ .
an = 2n + 1
Series –
the sum of a certain
number of terms of a sequence
n
Sigma Notation :
a
i 1
Stop
Formula
i
Start
“Add up the terms in the sequence
beginning at term number 1 and going
through term number “n”.
4
1. -5i 5 1 5 2 5 3 5 4 50
i 1
5
2.
i 1
7
1
2
3. i
i3
6
4.
3
i 1
25
i 1
2
25
Infinite Sequence
a1 , a2 , a3 , a4 ,..., ai ,...
Infinite Series
a1 a2 a3 a4 ... ai ... ai
i 1
Finite Series or nth Partial Sum
n
a1 a2 a3 a4 ... an ai
i 1
SUMMATION NOTATION
Sum of the terms of a finite sequence
Upper limit of summation
(Ending point)
n
a
i 1
i
Lower limit of summation
(Starting point)
Sequence
• There are 2 types of Sequences
Arithmetic:
You add a common difference each
time.
Geometric:
You multiply a common ratio each time.
Arithmetic Sequences
Example:
• {2, 5, 8, 11, 14, ...}
Add 3 each time
• {0, 4, 8, 12, 16, ...}
Add 4 each time
• {2, -1, -4, -7, -10, ...}
Add –3 each time
Arithmetic Sequences
• Find the 7th term of the sequence:
2,5,8,…
Determine the pattern:
Add 3 (known as the common difference)
Write the new sequence:
2,5,8,11,14,17,20
So the 7th number, or a7, is 20
Arithmetic Sequences
• When you want to find a large sequence,
this process is long and there is great room
for error.
• To find the 20th, 45th, etc. term use the
following formula:
an = a1 + (n - 1)d
Arithmetic Sequences
an = a1 + (n - 1)d
Where:
a1 is the first number in the sequence
n is the number of the term you are
looking for
d is the common difference
an is the value of the term you are
looking for
Arithmetic Sequences
• Find the 15th term of the sequence:
34, 23, 12,…
Using the formula an = a1 + (n - 1)d,
a1 = 34
d = -11
n = 15
an = 34 + (n-1)(-11) = -11n + 45
a15 = -11(15) + 45
a15 = -120
Arithmetic Sequences
Melanie is starting to train for a swim meet.
She begins by swimming 5 laps per day for
a week. Each week she plans to increase her
number of daily laps by 2. How many laps
per day will she swim during the 15th week
of training?
Arithmetic Sequences
• What do you know?
an = a1 + (n - 1)d
a1 = 5
d= 2
n= 15
t15 = ?
Arithmetic Sequences
• tn = t1 + (n - 1)d
• tn = 5 + (n - 1)2
• tn = 2n + 3
• t15 = 2(15) + 3
• t15 = 33
During the 15th week she will swim 33 laps per
day.
Arithmetic Series
• The sum of the terms in a sequence are
called a series.
• There are two methods used to find
arithmetic series:
Formula
Sigma Notation
Geometric Sequences
•
In geometric sequences, you multiply by a
common ratio each time.
•
1, 2, 4, 8, 16, ...
multiply by 2
27, 9, 3, 1, 1/3, ...
Divide by 3 which means multiply by 1/3
•
Geometric Sequences
• Find the 8th term of the sequence:
2,6,18,…
Determine the pattern:
Multiply by 3 (known as the common ratio)
Write the new sequence:
2,6,18,54,162,486,1458,4374
So the 8th term is 4374.
Geometric Sequences
• Again, use a formula to find large numbers.
•
an = a1 • (r)n-1
Geometric Sequences
• Find the 10th term of the sequence :
4,8,16,…
an = a1 • (r)n-1
• a1 = 4
• r=2
• n = 10
Geometric Sequences
an = a1 • (r)n-1
a10 = 4 • (2)10-1
a10 = 4 • (2)9
a10 = 4 • 512
a10 = 2048
Geometric Sequences
• Find the ninth term of a sequence if
a3 = 63 and r = -3
a1= ?
n= 9
r = -3
a9 = ?
There are 2 unknowns so you must…
Geometric Sequences
• First find t1.
• Use the sequences formula substituting t3 in for
tn. a3 = 63
• a3 = a1 • (-3)3-1
• 63 = a1 • (-3)2
• 63= a1 • 9
• 7 = a1