Transcript click here for nth term sequences

Use the distributive property to solve the
following problem: Remember to combine
like terms.
4 + 6(n -1)
3, 9, 27, 81,… What would the 5th number
be,
the 10th number,
the 73rd number?
Activator
The nth term
What is it?
The nth term is used to find any unknown
number in a pattern. For example. What is
the 50th number in this pattern?
2, 5, 8, 11, 14 …
1st term 2nd term
Do you really have time to figure this
out? Nah.
A sequence is an ordered list of numbers. Each
number in a sequence is called a term.
When the sequence follows a pattern, the terms
in the sequence are the output values of a
function, and the value of each number depends
on the number’s place in the list.
By "the nth term" of a sequence we mean an
expression that will allow us to calculate the
term that is in the nth position of the
sequence. For example consider the sequence


2, 4, 6, 8, 10,...
The pattern is easy to see. We are adding 2
to the previous number.
◦
◦
◦
◦
◦
◦
The first term is two.
The second term is 4.
The third term is 6
The fourth term is 8
The tenth term is …
the nineteenth term is ….

Arithmetic Sequence: When the value
changes by a common number added or
subtracted. The difference is called the
common difference.
There are two types of patterns
When to use:
Equation:
Arithmetic nth
term rules
Equation:
When to use:
Use when the pattern is
increasing or decreasing
by addition or subtraction.
A + d(n-1)
A = 1st term in pattern
D = the difference between each
term
N = the position in the pattern
you are trying to find.
Arithmetic
nth term
rules
You can use a variable such as n, to represent a
number’s position in a sequence.
n (position in the sequence)
1
2
3
4
y (value of term)
2
4
6
8
The nth term of a linear number sequence
(a sequence that goes up or down by the
same amount each time) can be found by
using the following mathematical formula:
nth term = a + d(n-1)
 a is the first number in the number
sequence
 d is the common difference (what you are
adding or taking from term to term).
 n is the number of the term you are
trying to find.

Step 3 nth term formula is
a + d(n-1)
Find the Nth term rule for each
sequence
N:
1
2
3
4
5
Term:
5
8
11
14
17
N:
1
2
3
4
5
Term:
5
6
7
8
9
N:
1
2
3
4
5
Term:
2
7
12
17
22
To write the nth term rule always start
with a+d(n-1) find the common
difference then simplify your
expression. This is the rule of your
sequence.
1. 6, 12, 18, 24,…
 2. 24, 21, 18, 15,…
 3. -28, -24, -20…
 4. 45, 42, 39…

Find the nth term expression
1. 1, 7, 13, 19, …
Find the 25th term
2. 3, 6, 9, 12, … Find the 10th term
3. 18, 14, 10, 6 … find the 30th term
4. -55, -50, -45, -40,… find 100th term
Find the nth term rule and then
the missing numbers.
How is finding the nth term different than
finding a function rule?
Assessment Prompt

From yesterday,

What does a, d, and n stand for?
Activator
In a geometric sequence, each term is
multiplied by the same amount to get the
next term in the sequence.
There are two types of patterns
A sequence where the numbers increase by
multiplication.
 Formula: nth term = a1(r n-1)
 r = common ratio (what is being multiplied)

n (position in the sequence)
1
2
y (value of term)
2
6
Geometric
Sequence
th
n-1
Find the 7 term: N = 2(3 )
3
4
18 54
When to use:
Equation:
Geometric nth
term rules
When to use:
Use when the pattern increases
or decreases through
multiplying or dividing.
Geometric
nth term
rules
Equation:
ar(n-1)
A = 1st term
R = the ratio between each number
N = term you are finding
1)
2)
3)
Find the Nth term rule for each
sequence
N:
1
2
3
4
5
Term:
5
10
20
40
80
N:
1
2
3
4
5
Term:
3
6
12
24
48
N:
1
2
3
4
5
Term:
4
8
16
32
64
1. 1, 4, 16, 64, …
Find the 12th term
2. 5, 15, 45, 135, … Find the 7th term
3. -4, -8, -16, -32 … find the 9th term
Find the nth term and then the
missing numbers.
When do you use the arithmetic rule?
Use the arithmetic rule when the sequence is
changing by addition or subtraction.
When do you use the geometric rule?
Use the geometric rule when the sequence is
changing by multiplication or division.
Assessment Prompt
Additional Example 1A: Identifying Patterns in a
Sequence
Tell whether the sequence of y-values is
arithmetic or geometric. Then find y when n = 5.
n
1
2
3
y
-1
-4
-16
4
5
-64 -256
In the sequence -1, -4, -16, -64,
number is multiplied by 4.
,…, each
-64 ● 4 = -256. Multiply the fourth number by 4.
The sequence is geometric. When n = 5, y = -256.
Additional Example 1B: Identifying Patterns in a
Sequence
Tell whether the sequence of y-values is
arithmetic or geometric. Then find y when n = 5.
n
1
2
3
4
5
y
51
46
41
36
31
In the sequence 51, 46, 41, 36,
each time.
36 + (-5) = 31.
,…, -5 is added
Add -5 to the fourth number.
The sequence is arithmetic. When n = 5, y = 31.
Check It Out: Example 1A
Tell whether the sequence of y-values is
arithmetic or geometric. Then find y when n = 5.
n
1
2
3
4
5
y
12
16
20
24
28
In the sequence 12, 16, 20, 24,
each time.
24 + 4 = 28.
,…, 4 is added
Add 4 to the fourth number.
The sequence is arithmetic. When n = 5, y = 28.
Check It Out: Example 1B
Tell whether the sequence of y-values is
arithmetic or geometric. Then find y when n = 5.
n
1
2
3
y
-1
-3
-9
4
5
-27 -81
In the sequence -1, -3, -9, -27,
number is multiplied by 3.
-27 ● 3 = -81.
,…, each
Multiply the fourth number by 3.
The sequence is geometric. When n = 5, y = -81.
Additional Example 2A: Identifying Functions in
Sequences
Write a function that describes the sequence.
3, 6, 9, 12,…
Make a function table.
n
Rule
y
1
1•3
3
2
2•3
6
3
3•3
9
4
4•3
12
Multiply n by 3.
The function y = 3n describes this sequence.
Additional Example 2B: Identifying Functions in
Sequences
Write a function that describes the sequence.
4, 7, 10, 13,…
Make a function table.
n
Rule
y
1
3(1) + 1
4
2
3(2) + 1
7
3
3(3) + 1 10
4
3(4) + 1 13
Multiply n by 3
and add 1.
The function y = 3n + 1describes this sequence.
Check It Out: Example 2A
Write a function that describes the sequence.
5, 6, 7, 8,…
Make a function table.
n
Rule
y
1
1+4
5
2
2+4
6
3
3+4
7
4
4+4
8
Add 4 to n.
The function y = 4 + n describes this sequence.
Additional Example 3: Using Functions to Extend
Sequences
Holli keeps a list showing her cumulative
earnings for walking her neighbor’s dog. She
recorded $1.25 the first time she walked the
dog, $2.50 the second time, $3.75 the third
time, and $5.00 the fourth time. Write a
function that describes the sequence, and
then use the function to predict her earnings
after 9 walks.
Write the number of walks she recorded; 1.25,
2.50, 3.75, 5.00.
Make a function table.
Additional Example 3 Continued
n
Rule
y
1
1 • 1.25 1.25
2
2 • 1.25 2.50
3
3 • 1.25 3.75
4
4 • 1.25 5.00
y = 1.25n
Multiply n by 1.25.
Write the function.
9 walks correspond to n = 9. When n = 9, y = 1.25 • 9
= 11.25. Holli would earn $11.25 after 9 walks.
Lesson Quiz: Part I
Tell whether each sequence of y-values is
arithmetic or geometric. Write a function that
describes each sequence, and then find y when n
= 5.
1. 6, 12, 24, 48,…
geometric; an = 6(2n-1); 96
2. –3, –2, –1, 0,…
arithmetic; an= n – 4; 1
3. 24, 21, 18, 15,…
arithmetic; an = 27 – 3n; 12
Lesson Quiz for Student Response Systems
1. Tell whether the given sequence of
y-values is arithmetic or geometric.
Identify a function that describes the
sequence, and then find y when n = 5.
4, 8, 12, 16, …
A. arithmetic; y = 4n; 20
B. geometric; y = 2n; 10
C. arithmetic; y = 4 + n; 9
D. arithmetic; y = 2n + 2; 12
Lesson Quiz for Student Response Systems
2. Tell whether the given sequence of
y-values is arithmetic or geometric.
Identify a function that describes the
sequence, and then find y when n = 5.
–5, –4, –3, –2, …
A. arithmetic; y = n + 6; 11
B. geometric; y = 2n – 6; 4
C. arithmetic; y = n – 6; –1
D. geometric; y = n; 5
Lesson Quiz for Student Response Systems
3. Tell whether the given sequence of
y-values is arithmetic or geometric.
Identify a function that describes the
sequence, and then find y when n = 5.
16, 12, 8, 4, …
A. arithmetic; y = 20 – 2n; 10
B. geometric; y = 4n; 20
C. arithmetic; y = 20 – 4n; 0
D. geometric; y = 30 – 6n; 0
1. 3n – 2
2. 5n
3. -2n + 1
4. 7n
Write the number sequence
Review: Give an
example of each:
natural #, whole #,
integer, rational, and
irrational
Identify the sequence
7,11,15,19
Find the 32nd term of
the sequence to the
left
Identify the sequence
500, 250,125,62.5
Review: Chris bought
a jet ski for $7500. He
Find the 8th term of the
sold it 4 years later for
sequence to the left
$2300. Find the % of
change
Identify the sequence
6,-12,24,-48
Find the 11th term of
the sequence to the
left
Summarizer:
Review: Identify as
direct or inverse and
the constant:
5
8
11
14
15
24
33
42