Transcript Sequences

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 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 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.
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
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
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
Ex. 2
Find the first four terms of the sequence
(1) n
an 
2n  1
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)