Sequences. Sequence There are 2 types of SequencesArithmetic: You add a common difference each time....
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Transcript of Sequences. Sequence There are 2 types of SequencesArithmetic: You add a common difference each time....
![Page 1: Sequences. Sequence There are 2 types of SequencesArithmetic: You add a common difference each time. Geometric: Geometric: You multiply a common ratio.](https://reader035.fdocuments.us/reader035/viewer/2022070407/56649e245503460f94b11fec/html5/thumbnails/1.jpg)
Sequences
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Sequence
There are 2 types of Sequences
Arithmetic:You add a common difference each time.
Geometric:You multiply a common ratio each time.
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Arithmetic SequencesExample:• {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
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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 is 20
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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
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Arithmetic Sequences
an = a1 + (n - 1)dWhere:
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
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Arithmetic Sequences• Find the 15th term of the sequence:
34, 23, 12,…
Using the formula an = a1 + (n - 1)d,
a1 = 34d = -11n = 15
an = 34 + (n-1)(-11) = -11n + 45
a15 = -11(15) + 45
a15 = -120
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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?
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Arithmetic Sequences• What do you know?
an = a1 + (n - 1)d
a1 = 5
d= 2
n= 15
t15 = ?
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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.
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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
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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:2,6,18,54,162,486,1458,4374So the 8th term is 4374.
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Geometric Sequences
• Again, use a formula to find large numbers.
• an = a1 • (r)n-1
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Geometric Sequences
• Find the 10th term of the sequence :
4,8,16,…
an = a1 • (r)n-1
• a1 = 4• r = 2• n = 10
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Geometric Sequences
an = a1 • (r)n-1
a10 = 4 • (2)10-1
a10 = 4 • (2)9
a10 = 4 • 512
a10 = 2048
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Geometric Sequences
• Find the ninth term of a sequence if a3 = 63 and r = -3a1= ?n= 9r = -3a9 = ?There are 2 unknowns so you must…
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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
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Geometric Sequences
• Now that you know t1, substitute again to find tn.
an = a1 • (r)n-1
a9 = 7 • (-3)9-1
a9 = 7 • (-3)8
a9 = 7 • 6561
a9 = 45927
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SequenceA sequence is a set of numbers
in a specific order
Infinite sequence
Finite sequence
,...,...,,,, 4321 naaaaa
naaaaa ,...,,,, 4321
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Sequences – sets of numbers
Notation:
represents the formula for finding terms
term numberna
n
Examples:
If 2 3, find the first 5 terms.na n
term. 20th the find ,13 If nan
th4
nd32
is the notation for the 4 term
is the notation for the 32 term
a
a
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Ex.1 Find the first four terms of the sequence
23 nan
12)1(31 a42 a
73 a
104 a
First term
Second term
Third term
Fourth term
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Ex. 2
12
)1(
n
an
n
Find the first four terms of the sequence
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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
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Writing Rules for Sequences
Try these!!!
3, 6, 9, 12, ___ , ___,……. _____ .
1/1, 1/3, 1/5, 1/7, ___ , ___,……. _____ .
an = 3n, an = 1/(2n-1)