Arithmetic Sequences
Identify the pattern in the sequence as:
arithmetic, geometric, or neither.
7, 11, 15, 19, … Answer: arithmetic
You added to generate each new term.
What is the rule used to generate new terms in the sequence?
Write it as a variable expression, and use n to represent the last number given.
7, 11, 15, 19, … Answer: n + 4 (since you add 4 to
b.
generate each new term)
What are the next 3 terms in the sequence?
7, 11, 15, 19, 23, 27, 31
circle
arithmetic geometric neither
Rule
17 , _____
9 , _____
1
Ex #2 49, 41, 33, 25 , _____
n + (−8)
_______________________
*Yes, this is just subtraction; however, since arithmetic means adding, write is as addition.
Geometric Sequences
Identify the pattern in the sequence as:
arithmetic, geometric, or neither.
3, 6, 12, 24, … Answer: geometric
You multiplied to generate each new term.
What is the rule used to generate new terms in the sequence?
Write it as a variable expression, and use n to represent the last number given.
3, 6, 12, 24, … Answer: 2n
b.
(since you multiplied by 2 to
generate each new term)
What are the next 3 terms in the sequence?
3, 6, 12, 24, 48 , 96 , 192
Ex #2
3
21
3
7203, 1029, 147, _____ , _____ , _____
7
circle
arithmetic geometric neither
Rule
1
n
_______________________
7
*Yes, this is just division; however, since geometric means multiply, write is as multiplication.
Other Sequences
Describe the pattern in the sequence and identify the sequence as
arithmetic, geometric, or neither.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
Answer: neither
There’s a pattern, but you’re neither adding nor multiplying by the same number.
What is the rule used to generate new terms in the sequence?
Since the pattern is neither arithmetic nor geometric, you can state the rule in words.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … Answer: You add the
last 2 terms
together to
b.
generate each new
term)
What are the next 3 terms in the sequence?
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 , 89 =
, 144 , 233 , 377
Negative Number Sequences
a. −37, −32, −27, −22, …
Is it arithmetic, geometric, or neither?
(…it’s rising slowly … signs not changing …)
What’s the rule?
n+5
arithmetic – you’re adding +5
b. −1, −7, −49, −343, …
List the next 3 terms.
−17, −12, −7, …
(…it’s falling quickly … signs not changing …)
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms.
geometric – you’re multiplying by 7
c. −99, −103, −107, −111
7n
−2401, −16,807 , −117,649
(…it’s falling slowly … signs not changing …)
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms.
arithmetic – you’re adding by −4
n + −4
−115, −119, −123
d.
−4 , 12 , −36 , 108 , …
Is it arithmetic, geometric, or neither?
geometric – you’re multiplying by −3
(…it’s rising quickly … signs alternating … )
What’s the rule? List the next 3 terms.
−3n
−324, 972, −2916
Decimal Number Sequences
a. 0.6 , 1.86 , 5.766 , 17.8746 (… number of decimal places increasing … )
(…divide 1st two terms, then the last 2 … same?)
Is it arithmetic, geometric, or neither?
geometric – you’re multiplying by 3.1
b. 4.7 , 7 , 9.3 , 11.6 , …
Is it arithmetic, geometric, or neither?
arithmetic – you’re adding 2.3
c. 4.5, 14.75, 25, 35.25,..
Is it arithmetic, geometric, or neither?
arithmetic – you’re adding 10.25
d. 1.6, 6.4, 25.6, 102.4,..
Is it arithmetic, geometric, or neither?
geometric – you’re multiplying by 4
What’s the rule?
3.1n
List the next 2 terms.
55.41126, 171.774906
(…divide 1st two terms, then the last 2 … same?)
(…subtract 1st two terms, then the last 2 … same?)
What’s the rule?
n + 2.3
List the next 2 terms.
13.9, 16.2, …
(…divide 1st two terms, then the last 2 … same?)
(…subtract 1st two terms, then the last 2 … same?)
What’s the rule?
n + 10.25
List the next 2 terms.
45.5, 55.75, …
(…subtract 1st two terms, then the last 2 … same?)
(…divide 1st two terms, then the last 2 … same?)
What’s the rule?
4n
List the next 2 terms.
409.6, 1638.4, …
Fractional Sequences
a.
1,●3 = 3 1,● 8 =8 13, 3,● 6 =18
8 ●?3 =24 3 ● 8 =24 24 4 ● 6? =24
? , ? , ? , ? ,…
8
3
24
4
Is it arithmetic, geometric, or neither?
arithmetic – you’re adding by 5
24
b.
1. Find the Least Common Denominator.
2. Rewrite each fraction with a new
numerator and denominator.
What’s the rule? List the next 3 terms.
n+ 5
24
28 ●16=, 448 7● 16,= 112 7 ● 4,=28 7 , …
1 ● 16? =16 1 ● 16=16 4 ● 4? =16 16
? ,
? ,
?
4
,
? ,…
16
Is it arithmetic, geometric, or neither?
23 , 28 , 33 or 23 , 7 , 11
24 24 24
24 6 8
1. Find the Least Common Denominator.
2. Rewrite each fraction with a new
numerator and denominator.
What’s the rule?
0.25n or 1n
4
4
geometric – you’re multiplying by 0.25, or 1
List the next 3 terms.
7 , 7 , 7
64
256 512
Practice with Sequences
0, 4.5, 9, 13.5, …
Is it arithmetic, geometric, or neither?
arithmetic
What’s the rule?
n + 4.5
‒3, ‒ 6, ‒12, ‒24, ‒48, . . .
List the next 3 terms
18, 22.5, 27
1, ‒3, 9, ‒27, 81, . . .
Is it arithmetic, geometric, or neither?
geometric
What’s the rule?
−3n
geometric
List the next 3 terms
−243, 729, −2187
List the next 3 terms
−1n
4, −4, 4
arithmetic
What’s the rule?
n + (−3)
geometric
List the next 3 terms
−96, −192,
−384
2n
Is it arithmetic, geometric, or neither?
neither
What’s the rule?
List the next 3 terms
add 1, then
add −1
2, 1, 2
0.5, 2.5, 4.5, 6.5, …
What’s the rule?
7, 4, 1, ‒2, ‒5, . . .
Is it arithmetic, geometric, or neither?
What’s the rule?
1, 2, 1, 2, 1, . . .
‒4, 4, ‒4, 4, ‒4, . . .
Is it arithmetic, geometric, or neither?
Is it arithmetic, geometric, or neither?
Is it arithmetic, geometric, or neither?
arithmetic
What’s the rule?
n+2
List the next 3 terms
8.5, 10.5, 12.5
‒5, 10, ‒20, 40, ‒80, . . .
List the next 3 terms
−8, −11, −14
Is it arithmetic, geometric, or neither?
geometric
What’s the rule?
−2n
List the next 3 terms
160, −320, 640
More Practice with Sequences
0, ‒2, ‒5, ‒9, ‒14, . . .
81, 27, 9, 3, 1, . . .
Is it arithmetic, geometric, or neither?
neither
What’s the rule?
add −2, then
add −3, then
−4, …
List the next 3 terms
−20, −27, −35
‒80, ‒76, ‒72, ‒68, ‒64, . . .
Is it arithmetic, geometric, or neither?
arithmetic
Is it arithmetic, geometric, or neither?
geometric
Is it arithmetic, geometric, or neither?
geometric
What’s the rule?
n+4
What’s the rule?
1
8
n
What’s the rule?
6n
−60, −56, −52
List the next 3 terms
5
4096
5
32768
List the next 3 terms
8
3
geometric
What’s the rule?
1n
3
List the next 3 terms
1
3
1
9
1
27
0.3, 0.6, 0.9, 1.2, …
List the next 3 terms
5
512
Is it arithmetic, geometric, or neither?
, 16, 96
Is it arithmetic, geometric, or neither?
arithmetic
What’s the rule?
n + 0.3
Is it arithmetic, geometric, or neither?
What’s the rule?
arithmetic
n+2
3
Is it arithmetic, geometric, or neither?
What’s the rule?
arithmetic
n+1
8
List the next 3 terms
1.5, 1.8, 2.1
List the next 3 terms
10
,
3
4,
14
3
List the next 3 terms
3
7
,
, 1
4
8
Functions
A Coke machine charges $1.00 for a soda.
~ If your input is 1 quarter, your output will be 0 sodas.
~ If your input is 2 quarters, your output will be 0 sodas.
~ If your input is 4 quarters, your output will be 1 soda.
Is the machine doing its function
correctly?
~ Later, you input 4 quarters, but the output is 2 sodas?
Is the machine doing its function
correctly?
A relation is a function when:
~ No inputs repeat.
or
~ If an input repeats, it’s always paired
with the same output.
1
2
4
4
0
0
1
2
Functions
Determine whether the relation is a function.
1. {(–3, –4), (–1, –5), (0, 6), (–3, 9), (2, 7)}
Answer: It is NOT a function (an x−value, −3, repeats with a different y−value)
2.
{(2, 5), (4, –8), (3, 1), (6, −8), (–7, –9)}
Answer: It IS a function (no x−values repeat)
3.
5.
6.
It is NOT a function (an x−value, 1,
repeats with a different y−value)
4.
It IS a function (no x−values repeat).
It IS a function
(an x−value, −4,
repeats with the
SAME
y−value, 11)
It IS a function
(no x−values
repeat)
Functions
Determine whether each graph is a function. Explain.
If NO x−values repeat, it IS a function.
Use “vertical line test” to test for a function:
If NO x−values repeat, it IS a function.
Use “vertical line test” to test for a function:
1. Hold a pencil vertically ...
1. Hold a pencil vertically ...
2. Then, slide it across the curve.
2. Then, slide it across the curve.
*Does the pencil ever hit the curve TWICE?
*Does the pencil ever hit the curve TWICE?
If the pencil hits
the curve ONCE,
it IS a function.
If the pencil hits the
curve TWICE, it is
NOT a function.
If the pencil hits
the curve ONCE,
it IS a function.
If the pencil hits the
curve TWICE, it is
NOT a function.
The pencil hits the curve ONCE, so it The pencil hits the curve TWICE, so it
PASSES the vertical line test.
FAILS the vertical line test.
It IS a function.
It is NOT a function.
Functions
There are different ways to show each part of a function. Let’s use the
example of: The effect of temperature on cricket chirps
This is the list of all input (x) values.
This is the list of all output (y) values.
Which variable causes
Which variable responds
the change?
to the change?
Which letter is listed first in an ordered
Which letter is listed second in an
pair?
ordered pair?
This is what goes in.
This is what comes out.
Conclusion: As temperature increases, cricket chirps increase.
(Summary):
Functions
A teacher displays the results of
her survey of her students.
texts per week
a. What is the input?
b. What is the independent variable?
texts per week
output?
average quiz score
dependent variable?
avg quiz score
c. What are all the x−values?
{10, 25, 100, 200}
y−values?
{81, 87, 94}
d. What’s the domain?
{10, 25, 100, 200}
range?
{81, 87, 94}
Writing Functions As Variable Expressions
Matt is a manager at Dominos. He earns a
salary of $500/week, but he also gets $0.75
for every pizza he sells.
Write a variable expression you could use to
find his total weekly pay.
Ned sells tandem skydives. He makes
$1000 for a full plane of jumpers, but he
has to pay the pilot $25 per jumper.
Write a variable expression you could use
to find his total pay for every full plane.
salary + pay per pizza = total weekly pay
Ned’s pay ‒ pay per jumper = total pay
‒ 25 • j
500 + 0.75 • p
1000
500 + 0.75p
1000 ‒ 25j
Adam drives a truck, and his mileage
chart is above.
Write a variable expression you could
use to find his total amount of gas he
has in his tank?
gas he started with – gas per mile = gas remaining
35.1
− 0.6 • m
35.1 ‒
0.6m
Lambert is
running a food
donation drive,
and the results
are to the right.
Write a variable
expression you
could use to find
his total pounds
of food
donated?
starting food + food per day
1
97 3
• d
1
+ 26
+
d
Completing a Function Table
To graph a function
~ Step 1: Pick a value for x ( I recommend “0”), then ...
* Write “0” under “x”, ...
* ... re−write your equation, then plug in “0” for x, then ...
* ... plug in “0” for the x−value of the ordered pair.
~ Step 2: To figure out the y−value,
* Use order of operations to evaluate the expression. The “answer” is your y−value, so ...
> write it under “y”, ...
> ... then plug it in for the y−value of your ordered pair.
x
y = 4x + 3
0
1
y = 4(
y = 4(
2
y = 4(
0)+3
1)+3
2)+3
y
(x,y)
3 ( 0 , 3)
7 ( 1, 7 )
11 ( 2, 11 )
Graphing Functions with Ordered Pairs
Plot all three ordered pairs
from your function table
If they all line up,
~ get a ruler, then ...
~ draw a straight line
through all 3 points.
If they don’t line up,
~ choose a new x−value
~ plug it in your
function table
~ plot your new point
(hopefully, they line up)
16
Completing a Function Table
To graph a function
~ Step 1: Pick a value for x ( I recommend “0”), then ...
* Write “0” under “x”, ...
* ... re−write your equation, then plug in “0” for x, then ...
* ... plug in “0” for the x−value of the ordered pair.
~ Step 2: To figure out the y−value,
* Use order of operations to evaluate the expression. The “answer” is your y−value, so ...
> write it under “y”, ...
> ... then plug it in for the y−value of your ordered pair.
x
0
4
8
y=
1
4
x – 2
1
4(
1
4(
0 )–
y=
4)–
1
y =4( 8 ) –
y=
2
2
2
y
(x,y)
–2 ( 0 ,–2)
–1 ( 4 ,–1 )
0 ( 8, 0 )
Graphing Functions with Ordered Pairs
Plot all three ordered pairs
from your function table
If they all line up,
~ get a ruler, then ...
~ draw a straight line
through all 3 points.
If they don’t line up,
~ choose a new x−value
~ plug it in your
function table
~ plot your new point
(hopefully, they line up)
18
Graphing Horizontal (y =) Lines.
Graph y = 4
~ Write an ordered pair
with any x−value.
( 0 , 4)
~ The y−value is 4.
* Why? Because the
original equation is
y = 4.
~ Pick another x−value.
The y−value will be 4.
( 1, 4 )
(2, 4 )
~ Plot the points, then
draw your line.
19
Graphing Vertical (x = ) Lines.
Graph x = –7
~ Write an ordered pair
with any y−value.
(–7, 0 )
~ The x−value is –7.
* Why? Because the
original equation is
x = –7.
~ Pick another y−value.
The x−value is –7.
(–7 , 1)
(–7 , 2)
~ Plot the points, then
draw your line.
20
In an arithmetic
sequence, you add
the ________ to
get each new term.
common
difference
14, 3, −8, ...
+(−11) +(−11)
In a geometric
sequence, you
multiply by the
________ to get
each new term.
common
ratio
3, 21, 147,...
•7 •7
30
31
32
33
34
6n , geometric
7n , geometric
n + 1.1 , arithmetic
n + 12 , arithmetic
3n , geometric
•3, •4, •5 … , neither
20.6, 24.6, 28.6
53, 58, 63
Sequences
1.3, 1.6, 1.9
204.8, 1638.4, 13107.2
768, 3072, 12288
1
5
2013, 2020, 2037
,
1
25
,
1
125