sequences. sequence there are 2 types of sequencesarithmetic: you add a common difference each time....
TRANSCRIPT
Sequences
Sequence
There are 2 types of Sequences
Arithmetic:You add a common difference each time.
Geometric:You multiply a common ratio each time.
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
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
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)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
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
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.
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:2,6,18,54,162,486,1458,4374So 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 = -3a1= ?n= 9r = -3a9 = ?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
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
SequenceA sequence is a set of numbers
in a specific order
Infinite sequence
Finite sequence
,...,...,,,, 4321 naaaaa
naaaaa ,...,,,, 4321
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
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
Ex. 2
12
)1(
n
an
n
Find the first four terms of the sequence
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
Writing Rules for Sequences
Try these!!!
3, 6, 9, 12, ___ , ___,……. _____ .
1/1, 1/3, 1/5, 1/7, ___ , ___,……. _____ .
an = 3n, an = 1/(2n-1)