Generic symbol for a
TERM in a sequence a n a
1
= 1 st term a
30
= 30 th term a n
= n th term
Example 2:
Find the next 3 terms of the given sequence
Example 1:
Find the first five terms of the given sequence: a n

5 n

3
6, 1, -4, -9, …
An arithmetic sequence is a sequence where the difference between consecutive terms is constant. The difference between the consecutive terms is called the common difference. (d)
Example 1:
Is the sequence an arithmetic sequence?
If so, state the common difference.
A) 1, 4, 7, 10, … B) 2, 4, 8, 16, …
Example:
4, 7, 10, 13,… a
1
= _____ d = _____
C) 1, -5, -11, -17, …
1

Example 2:
Find the next 3 terms of the arithmetic sequence.
B) -5, -1, 3, … A) 12, 17, 22, … C) 12, -3, -18, …
WHAT IF YOU WERE ASKED:
What is the 11 th term of the following arithmetic sequence? 6, 13, 20, …
TWO APPROACHES:
1. RECURSIVE: Continue the pattern until you reach the 11 th number.
2. EXPLICIT: Use a formula that represents the pattern
Example 3: Find a Specific Term
A) What is the 11 th term of the arithmetic sequence 6, 13, 20, … ?
Arithmetic Sequence Formula a n
= a
1
+ (n –1)d a
1
= 1 st term a n
= n th term n = subscript of a n
(the counting number of the term) d = the common difference
B) What is the 46 th term of the arithmetic sequence C) What is the 110 th term of the arithmetic sequence
3, 5, 7, … ? -5, -9, -13, …. ?
2

Example 4: Find a specific formula (pattern) Write a formula to represent the given sequence.
A) 2, 5, 8, 11, …. B) 10, 6, 2, -2, …
Example 5: Finding missing terms
HINT: Which variable do you need to know?
Find the missing numbers in each arithmetic sequence.
a n
= a
1
+ (n –1)d
A) 80, ____, ____, 125, …. B) 146, ____, _____, ____, 78, …. C) 35, _____, 53 , …
Arithmetic Mean
:
In an arithmetic sequence, the middle term of any three consecutive terms is the arithmetic mean (the average) of the other two terms.
The number between
Careful: This doesn’t work if you have more than 1 missing term! See example 5A and 5B! x and y is x
 y
2
Example 6:
A) Find the missing number 35, _____, 53 , …
C) The 9 th and 11 th terms of an arithmetic sequence
are 132 and 98, respectively. What is the 10 th term?
B) Given that a
5
= 15 and a
7
= 59, find a
6.
D) Find the missing number 15, _____, 27
3

An geometric sequence is a sequence where the ratio between consecutive terms is constant. The ratio between the consecutive terms is called the common ratio. (r)
Example 1:
Is the sequence a geometric sequence?
If so, state the common ratio, r.
A) 3, 12, 48, … B) 16, 24, 36, …
D) 5, 10, 50, …. E) -8, 4, -2, 1, ….
Example 2:
Find the next 3 terms of the geometric sequence.
A) 15, 30, 60, … B) -120, 30, -7.5, …
WHAT IF YOU WERE ASKED:
What is the 8 th term of the following geometric sequence?
TWO APPROACHES:
1. RECURSIVE: Continue the pattern until you reach the 8 th number.
Example:
4, 12, 36, 108… a
1
= _____ r = _____
C) 3, 6, 9, …
F)
1 , 1 , ,...
81 27 9
C) 12, 18, 27, …
6, 12, 24, …
4

2. EXPLICIT: Use a formula that represents the pattern
Example 3: Find a Specific Term
A) What is the 8 th term of the arithmetic sequence 6, 12, 24, … ?
Arithmetic Sequence Formula n

1 a n
 
1 a
1
= 1 st term a n
= n th term n = subscript of a n
(the counting number of the term) r = the common ratio
B) What is the 10 th term of the geometric sequence C) What is the 7 th term of the geometric sequence
-36, 18, -9, …. ? 4, 12, 36, … ?
D) What is the 8 th term of a geometric sequence for which a
1
 
3 and r
 
2 ?
Example 4: Find a specific formula (pattern)
Write a formula to represent the given sequence.
A) 2, 6, 18, …. B) 10, 2, 0.4, … a n
  n

1
5

Example 5: Finding missing terms
HINT: Which variable do you need to know?
Find the missing numbers in each geometric sequence.
a n
  n

1
A) 2, ____, ____, -54, …. B) 9, ____, _____, ____, 144, …. C) 28, _____, 7 , …
Geometric Mean
:
In a geometric sequence, the middle term of any three consecutive terms is the geometric mean of the other two terms.
The number between
Careful: This doesn’t work if you have more than 1 missing term! See example 5A and 5B! x and y is
 xy
Example 6:
A) Find the missing number 28, _____, 7 , …
C) The 9 th and 11 th terms of a geometric sequence
are -8 and -2, respectively. What is the 10 th term?
B) Given that a
5
= 5 and a
7
= 2.8125, find a
6.
D) Find the missing number 16, _____, 9
6

An arithmetic series is a sum of the terms in an arithmetic sequence (see lesson 9.2)
Example 1:
Find the sum of the first 100 positive integers.
Example 2: Finding Finite Sums
A) What is the sum of the arithmetic series where
a
1
= 7 , a n
= 79, and n = 8 ?
B) What is the sum of the arithmetic series where
a n
= 80, n=11, and d = 7 ?
C) What is the sum of the arithmetic series where
14 + 17 + 20 + … + 116 ?
FINITE Arithmetic Series Formula
S n
 n  a
1
 a n

2 a
1
= 1 st term a n
= n th term (the last term in the series) n = subscript of a n
(the counting number of the term)
S n
= the Sum of the n terms in the series
9.2 Arithmetic Sequence: a n
= a
1
+ (n –1)d
D) What is the sum of the arithmetic series where
20 + 18 + 16 + … + -24 ?
7

B)
Summation Notation
You can use the Greek capital letter sigma Σ to indicate a sum. With it, you use limits to indicate how many terms you are adding. Limits are the least and greatest values of n in the series. You write the limits below and above the Σ to indicate the first and last terms of the series. last value of n n
10 

1
3 n first value of n formula for the terms in the series
Write out this arithmetic series and find the sum.
Example 3: Finding Sum from Summation Notation
What is the sum of the given series?
A) n
40 

1
( 3 n

8 )
Strategy:

Find a
1
(the 1 st term)
Plug the lower limit into the formula for the nth term.

Find a n
(the last term)
Plug the upper limit into the formula for the nth term.

Find n (the number of terms) n = upper limit – lower limit + 1

Find the SUM n
Use S n
  a
1
 a n

2 n
50 
 1
( 4 n

7 )
C) n
12 
 4
(

2 n )
8

Example 4: Vocabulary Review
Draw a line to match the word/phrase in column A with the correct definition in column B.
9

A geometric series is a sum of the terms in a geometric sequence (see lesson 9.3)
Example 1: (from your textbook pg 596)
According to the story, what would the first 5 terms of this series be?
Is this an arithmetic series or a geometric series? How can you tell?
Use the Geometric Series formula at the right.
How many kernels of wheat did the soldier request?
FINITE Geometric Series Formula
S n
 a
1
 r n
(1 )
1
 r
or S n a
1
= 1 st term a n
= n th term (the last term in the series) n = subscript of a n
(the counting number of the term) r = the common ratio
S n
= the Sum of the n terms in the series
Example 2: Sums of Finite Geometric Series
Find the sum of the finite geometric series with the following information
A) a
1
= -15, r = -2, and n = 6 B ) a
1
=81, r = 1
3
, n = 5
C) 4 + 12 + 36 +…+ 2916 D) –6 + 18 –54 + …+ 13122
 a
1
 a r n
1
 r
10

Infinite Geometric Series
Think about the following infinite geometric series:
4 1
1 1

1

1
4 16 64 256

...
What are the following “Partial Sums”?
S
1
=
4
S
6
=
4 1
1 1

1

1
4 16 64 256
S
2
=
4 1
S
7
=
4 1
1 1

1

1

1
4 16 64 256 1024
S
3
=
 
1
4
S
4
=   
1
4 16
S
5
=   
1

1
4 16 64
S
S
S
8
9
=
= n
4 1
4 1

1 1

1

1

1

1
4 16 64 256 1024 4096
1 1

1

1

1

1

1
4 16 64 256 1024 4096 16384
What appears to be happening?
In an infinite geometric series where r

1 , one of two things can happen:
Case 1: r

1 or r
 
1
Case 2:
1 r 1
Terms GROW rapidly.
Sum grows RAPIDLY.
S approaches

or

.
“DIVERGES”
Terms DIMINISH rapidly.
Sum starts growing by negligible amounts.
S approaches a finite sum. (an actual number!)
“CONVERGES”
11

Example 3: Sums of Infinite Geometric Series
Does the infinite geometric series converge or diverge?
If it converges, find the sum.
A) 18 9 4.5 ...
B) 18 6 2 ....
INFINITE Geometric Series Formula
S

1 a
1
 r a
1
= 1 st term r = the common ratio
S = the Sum of the series
C) 1
 
25

....
4 16
D) 81 27 9 ...
E)
 
Example 4: Repeating Decimals  Fractions
Think about
0.52
r 1
12