Transcript n - 1

GEOMETRIC
SEQUENCES
These are sequences where the ratio of
successive terms of a sequence is always
the same number.
This number is called the common ratio.
Geometric sequence
• A geometric sequence, in math, is a sequence
of a set of numbers that follow a pattern. We
call each number in the sequence a term.
• For examples, the following are sequences:
2, 4, 8, 16, 32, 64, .......
•
243, 81, 27, 9, 3, 1, ...............
• A geometric sequence is a sequence where
each term is found by multiplying or dividing
the same value from one term to the next. We
call this value "common ratio"
Looking at 2, 4, 8, 16, 32, 64, ......., carefully
helps us to make the following observation:
• As you can see, each term is found by multiplying 2, a common
ratio to the previous term
• add 2 to the first term to get the second term,
• but we have to add 4 to the second term to get 8.
• This shows indeed that this sequence is not created by adding or
subtracting a common term
Looking at 243, 81, 27, 9, 3, 1, ...............carefully
helps us to make the following observation:
• This time, to find each term, we divide by 3, a common
ratio, from the previous term
• Many geometric sequences can me modeled with an
exponential function
• an exponential function is a function of the form an where a
is the common ratio
Here is the trick!
2, 4, 8, 16, 32, 64, .......
• Let n represent any term number in the sequence
Observe that the terms of the sequence can be written as
21, 22, 23, ...
• We can therefore model the sequence with the following
formula: 2n
• Check:
• When n = 1, which represents the first term, we get 21 = 2
• When n = 2, which represents the second term, we get
22 = 2 × 2 = 4
Let us try to model 243, 81, 27, 9, 3, 1,
...............
• Common ratio? … divide by 3
• Let n represent any term number in the
sequence
• Observe that the terms of the sequence can
be written as 35, 34, 33, ...
• We just have to model the sequence: 5, 4, 3,
.....
243, 81, 27, 9, 3, 1, ...............
• Use arithmetic sequence to model 5, 4, 3, …
– Common difference -1
• The process will be briefly explained here
• The number we subtract to each term is 1
• The number that comes right before 5 in the
sequence is 6
• We can therefore model the sequence with the
following formula:
-1* n + 6
243, 81, 27, 9, 3, 1, ...............
• We can therefore model 243, 81, 27, 9, 3, 1, .........
with the exponential function below
3-n + 6
Check:
When n = 1, which represents the first term, we get
3-1 + 6= 35 = 243
When n = 2, which represents the second term, we
get
3-2 + 6= 34 = 81
Numerical Sequences and Patterns
Arithmetic Sequence
Add a fixed number to the previous term
Find the common difference between the previous & next term
Find the next 3 terms in the arithmetic sequence.
2,
5,
8,
11,
___,
14
+3the common
+3
+3
+3
+3
What is
Does thethe
same
difference between
difference
first and second
term?hold for
the next two terms?
___,
17
+3
___
21
Arithmetic Sequence
What are the next 3 terms in the arithmetic sequence?
17,
13,
9,
5,
___,
1
___,
-3
___
-7
An arithmetic sequence can be modeled using a function rule.
What is the common difference of the terms in
-4
the preceding problem?
Let n = the term number
Let A(n) = the value of the nth term
in the sequence
A(1) = 17
A(2) = 17 + (-4)
Term #
1
2
3
4
n
A(3) = 17 + (-4) + (-4)
Term
17 13
9
5
A(4) = 17 + (-4) + (-4) + (-4)
Relate
Formula
A(n) = 17 + (n – 1)(-4)
Arithmetic Sequence Rule
nth
term
first
term
term
number
Common
difference
Find the first, fifth, and tenth term of the sequence:
A(n) = 2 + (n - 1)(3)
First Term
A(n) = 2 + (n - 1)(3)
A(1) = 2 + (1 - 1)(3)
= 2 + (0)(3)
=2
Fifth Term
Tenth Term
A(n) = 2 + (n - 1)(3) A(n) = 2 + (n - 1)(3)
A(5) = 2 + (5 - 1)(3) A(10) = 2 + (10 - 1)(3)
= 2 + (4)(3)
= 2 + (9)(3)
= 14
= 29
Real-world and Arithmetic Sequence
In 1995, first class postage rates were raised to 32
cents for the first ounce and 23 cents for each
additional ounce. Write a function rule to model the
situation.
Weight (oz)
A(1)
Postage (cents) .32 + 23
A(2)
.32+.23+.23
A(3)
.32+.23+.23+.23
What is the function rule?
A(n) = .32 + (n – 1)(.23)
What is the cost to mail a 10 ounce letter?
A(10) = .32 + (10 – 1)(.23)
= .32 + (9)(.23)
= 2.39
The cost is $2.39.
n
Numerical Sequences and Patterns
Geometric Sequence
• Multiply by a fixed number to the previous term
• The fixed number is the common ratio
Find the common ratio and the next 3 terms in the sequence.
3,
12,
48,
192,
___,
768 _____,
3072 12,288
______
x4
x4
x4
x4
Does the same
What is the common
RATIO
RATIO between
thehold for the
next term?
two terms?
first and second
x4
x4
Geometric Sequence
What are the next 2 terms in the geometric sequence?
5
5
5
80,
20,
5, 4 , ___,
___64
16
An geometric sequence can be modeled using a function rule.
What is the common ratio of the terms in the 1
preceding problem?
4
Let n = the term number
Let A(n) = the value of the nth term
in the sequence
A(1) = 80
A(2) = 80 · (¼)
Term #
1
2
3
4
n
A(3) = 80 · (¼) · (¼)
5
Term
80 20
5
A(4) = 80 · (¼) · (¼) · (¼)
4
Relate
Formula
A(n) = 80 · (¼)n-1
Geometric Sequence Rule
nth
term
first
term
common
ratio
Term
number
Find the first, fifth, and tenth term of the sequence:
A(n) = 2 · 3n - 1
First Term
A(n) = 2· 3n - 1
Fifth Term
A(n) = 2 · 3n - 1
Tenth Term
A(n) = 2· 3n - 1
A(1) = 2· 31 - 1
A(5) = 2 · 35 - 1
A(10) = 2· 310 - 1
A(1) = 2
A(5) = 162
A(10) = 39,366
Hand Shake Problem:
If each member of this class shook hands with everyone
else, how many handshakes were there altogether?
People in class =
Mathematical Model:
Point = 1 person
Segment = handshake
Person
Point
(Term)
1
Handshakes 0
Segments
2
1
3
3
4
6
5 …
10
n
n ( n  1)
2
…
25
300
• What are Triangular Numbers?
These are the first 100 triangular numbers:
• The sequence of the triangular numbers
comes from the natural numbers (and zero), if
you always add the next number:
1
1+2=3
(1+2)+3=6
(1+2+3)+4=10
(1+2+3+4)+5=15
...
• You can illustrate the name triangular number
by the following drawing:
Triangular Number Sequence
This is the Triangular Number Sequence:
This sequence is generated from a pattern of dots which
form a triangle.
By adding another row of dots and counting all the dots
we can find the next number of the sequence:
1, 3, 6, 10, 15, 21, 28, 36, 45, ...
• We can make a "Rule" so we can calculate any
triangular number.
• First, rearrange the dots (and give each pattern a
number n), like this:
• Then double the number of dots, and form them
into a rectangle:
• The rectangles are n high and n+1 wide (and
remember we doubled the dots), and xn is
how many dots (the value of the Triangular
Number n):
• 2xn = n(n+1)
• xn = n(n+1)/2
Rule: xn = n(n+1)/2
• Example: the 5th Triangular Number is
• x5 = 5(5+1)/2 = 15
• Example: the 60th is
• x60 = 60(60+1)/2 = 1830
• Wasn't it much easier to use the formula than
to add up all those dots?