Download What is an arithmetic sequence?

Do Now
 Find the pattern and use it to find the next 3
numbers in the sequence:
1.
1, 3, 5, 7, 9, 11,___, ___, ___
2. 2, 7, 12, 17, 22, 27,___, ___, ___
3. 100, 97, 94, 91, 88, 85,___, ___, __
Arithmetic and
Geometric
Sequences
Arithmetic Sequences
Every day a radio station asks a
question for a prize of $150.
If the 5th caller does not
answer correctly, the prize
money increased by $150 each
day until someone correctly
answers their question.
Arithmetic Sequences
Make a list of the prize
amounts for a week
(Mon - Fri) if the contest
starts on Monday and no one
answers correctly all week.
Arithmetic Sequences
Monday :
Tuesday:
Wednesday:
Thursday:
Friday:
$150
$300
$450
$600
$750
Arithmetic Sequences
These prize amounts form a
sequence, more specifically
each amount is a term in an
arithmetic sequence. To
find the next term we just
add $150.
What is an arithmetic sequence?
A sequence in which
each term is found by
ADDING the same
number to the previous
term.
+4
4,
+4
+4
+4
8, 12 , 16, 20…………..
What is the common difference?
The difference between
each number. This
determines what is added to
each previous number to
obtain the next number.
4, 8, 12, 16, 20…………..
4 is the common difference
Arithmetic Sequences
 An arithmetic sequence is a set of numbers put




into a specific order by a pattern of addition or
subtraction.
an = a1 + (n – 1)d– This is the formula.
an represents the nth term, the unknown term
that you are trying to find, of a sequence.
a1 is the first term in a sequence.
n is an unknown term that is always the same
number as the n term in an.
Arithmetic Sequences (continued)
 The d in the formula is the
Common Difference between
each of the terms in a series.
 For example: 1, 5, 9, 13… The common difference
(d) is +4.
 The d term can also be negative:
10, 7, 4, 1, -2… The d term is -3
(this means that instead of
adding a number you
subtract it.)
Try These! Are they arithmetic sequences?
 Find the pattern and use it to find the next 3
numbers in the sequence:
1.
1, 3, 5, 7, 9, 11,___, ___, ___
2. 2, 7, 12, 17, 22, 27,___, ___, ___
3. 100, 97, 94, 91, 88, 85,___, ___, __
What is a geometric sequence?
A sequence in which
each term can be found
by multiplying the
previous term by the
same number.
x3
3,
x3
x3
x3
9, 27 , 81, 243…………..
What is the common ratio?
The number used to multiply
by each previous number to
obtain the next number.
2, 8, 32, 128, 512…………..
4 is the common ratio
Geometric Sequence
What if your pay check started
at $100 a week and doubled
every week. What would your
salary be after four weeks?
GeometricSequence
 Starting $100.
 After one week - $200
 After two weeks - $400
 After three weeks - $800
 After four weeks - $1600.
 These values form a geometric
sequence.
Geometric Sequences
 an = a1rn-1 Geometric Sequence formula.
 an is the unknown term (just like the arithmetic
sequences)
 a1 is the first term.
 r is the rate, also known as the common ratio. It
is the change between two terms in a geometric
sequence. It is either a number being multiplied
or divided. You can also multiply by (1) over the
number being multiplied.
More Geometric Sequences
 Some examples of geometric sequences are:
 1, 2, 4, 8, 16, 32…-- r = 2
 100, 50, 25, 12.5, 6.25…-- r = 1/2 (divide the
preceding number by 2.)
an=a1rn-1
How this relates to Real Life Outside Math Class
 A painter is a job that requires the use of an
arithmetic sequence to correctly space the things
he is painting. If the painter was painting stripes
on a wall, he could find the places to put the stripes
to evenly space them.
Another Real Life Slide
 If an owner of a store needed to count up the
amount of stuff they sell, or how much money
they make, he could use and arithmetic or
geometric sequence.
 If the owner had a pattern of how much money
they make as time progresses, that is a sequence.
The owner also needs these sequences if he/she
wants to predict the earnings of his or her store
in years to come.
Let’s Practice!
Is this an arithmetic or geometric
sequence?
10, 15, 20, 25, 30……
Arithmetic Sequence
+5
5 is the common difference
Is this an arithmetic or geometric
sequence?
2, 12, 72, 432, 2,592……
Geometric Sequence
x6
6 is the common ratio
Is this an arithmetic or geometric
sequence?
What is the common ratio or difference?
What is the next term in this
sequence?
-6
5,
-6
-1,
-6
-7
-6
-19
-13 , _____
Is this an arithmetic or geometric
sequence?
What is the common ratio or difference?
What is the next term in this
sequence?
+20
+20
+20
+20
-400,-380,-360,-340
-320
, ____
Is this an arithmetic or geometric
sequence?
What is the common ratio or difference?
What is the next term in this
sequence?
x-4
x-4
x-4
x-4
3,072
12, -48, 192 , -768____
Some Interesting Example Equations
Geometric example: find the nth term.
a1 = -10, r=4, n=2
an = -10(4)2-1
an = -10(4)1
an = -40
Arithmetic example: find a14, a1=4, d=6
a14= 4 + (14-1)6
a14= 4 + 78
a14= 82