Name______________________
Period_________
Unit 10: Sequences and Series
& Equation
DAY
1
2
3
4
5
6
7
TOPIC
INTRO TO SEQUENCES
SEQUENCES, SERIES AND SUMMATION
NOTATION
ARITHMETIC SEQUENCES
ARITHMETIC SERIES
QUIZ
GEOMETRIC SEQUENCES and
GEOMETRIC MEAN
FINITE and INFINITE GEOMETRIC SERIES
8
REVIEW OF GEOMETRIC SEQ./SERIES
Small QUIZ
9
10
REVIEW
TEST
ASSIGNMENT
P. 865 #1-10, 14, 15
12.2, pg. 874: 1, 2, 4, 6, 8, 14, 19, 36,
37, 40
12.3, pg. 884: 2-16 EVEN
12.3, pg. 884: 17-20, 33-35, 41-43
TBD
12.4, pg. 895: 2-13
12.4, pg. 895: 14-17 and
12.5, pg. 904: 1-7
BEGIN REVIEW-TEXT REVIEW
SECTION -12.2-12.5
p.913-915 18-62 (Every other Even)
REVIEW SHEET
We will have mini quizzes throughout the remainder of the year
in addition to normal quizzes/tests.
Page 1 of 18
Day 1: Intro to Sequences
& Equation
A ______________________ is an ordered set of numbers.
Example: 1, 3, 5, 7, 9, 11, …
What is the 1st term?
Each number is called a _________________ of the sequence.
3rd term?
6th term?
10th term?
The sequence above is called an _______________ sequence because it goes on forever (notice the …).
If the sequence does end (like 2, 4, 6, 8), then it is called a _______________ sequence.
Sometimes it is helpful to list your sequence in a table to help you map number of the term to the term itself.
Term Number
Term Value
Normally, we will use sequence notation: n and an .
n represents _____________________ and an represents _____________________
So for example, a4  7 means “The _________ term is equal to ________”
Examples: Use the sequence  2, 6, 10, 14, 18, 22, 26
a) a2  _____
b) a5  _____
c) a___  10
d) a___  14
d) a7  _____
Ok, now that we understand the notation, let’s talk a little more abstractly about sequences.
 What is the term previous to an ? __________.
 What is the next term after an ? ___________.
 What is the next term after an 3 ? ___________.
 What is the term previous to an 3 ? ___________.
A ____________________ ___________________ is a rule in which one or more previous terms are used to
generate the next term in the sequence.
Example: Start with the number 5, and then add 3 each time.
Sequence Notation:
Generate the first 5 terms:
Page 2 of 18
Example together: a1 = 5 and an  2(an1 )  3 , find the first 5 terms in the sequence.
On your own:
1. Find the first 5 terms in the sequence with a1 = - 2 and an  3(an1 )  2
2. Find the next 4 terms in the sequence which has a3  7 and an  2(an1 )  4
Question: What if I asked you to find the 100th term of the sequence, or the 1000th?
Answer: We must find an _________________________ formula.
An explicit formula defines the nth term in a sequence as a function of n.
Example together: find the first 5 terms of the sequence an  2n  3 .
On your own:
1. Find the first 5 terms of the sequence an  n 2  2n
2. Find the first 5 terms of the sequence an  3n  5 .
3. Find the 3rd, 6th, 10th and 17th terms in the sequence an  3n  1
Page 3 of 18
Applications:
1. Dave earns $10,000 for his first year of work, but will earn a 10% pay increase for each. How much
will he be making in year 5?
2. Continue the pattern below. Find the number of squares in the 5th term. How about the 20th term?
Extension (HW Questions):
Day 2: Sequence, Series, and Summation Notation
& Equation
Yesterday we looked at sequences like  3, 6, 9, 12, 15
Today we will look at a _______________, which is simply a sequence when you __________ up
all of the terms.
Just like a sequence, a series can be ___________________ (go on forever), or ________________.
We use the notation S n to represent the partial sum of a series.
Example: Use the series 5, 7,9,11,... to answer the following questions.
S4  ______
S2  ______
S _____  21
a5  ______
a_____  11
Page 4 of 18
Mathematicians are lazy  they always try to write short hand.
Short hand for series is using __________________ notation with the Greek letter Sigma (  ).
Ending Term #
5
2
i

Take the starting term # and plug it into the rule
 The value you get is a" firstTerm"
Repeat the process…
The “Rule”
i 1
until you get the ending term.
Remember, S n is the partial sum – we must add!
Starting Term #
Example #1:
Starting Term #
n
1
Ending Term #
2
3
4
5
an
Sn
Expand (find the terms) and evaluate (find the sum of the terms) each series below.
4
1.
5
 2n  1
2.
(1)k

k 3 3k
4.
k 1
)
k 1
n 1
7
3.
 (5)(2
6
n
2
 10
n 1
Page 5 of 18
Sometimes you need to work backwards…
Write the following series in summation notation
1) 3 + 5 + 7 + 9
2)
Hint: Write the # of the term above the term value
1 1 1 1
  
4 6 8 10
Def’n:
Write each series below in summation notation.
1.) 3  6  9 12 15
2.)
1 1 1 1 1
1
    
2 4 8 16 32 64
Some shortcuts for finding a series; If the series is …
n
a) constant,
then
 c  nc for example,
k 1
7
 3  7(3)  21
k 1
100
n  n  1
100 100  1
for example,  k 
 5050
2
2
k 1
k 1
(this is the sum of the first 100 counting numbers)
n
b) linear
then
c) quadratic
then
k 
n
k2 
k 1
n  n  1 (2n  1)
for example
6
n4
k
2

k 1
4  4  1 (2(4)  1)
6
Evaluate each series.
10
6
k 4
8
k
k 1
12
k
2
k 1
Page 6 of 18
Day 3: Arithmetic Sequences
& Equation
Warm-up: Describe the two sequences below
a) 5,8,11,14,17, 20, 23,...
b) 3, 6,12, 24, 48,96,...
The sequences above are known as (a)___________________ & (b)___________________.
Today we will focus on arithmetic sequences: A sequence in which all consecutive terms differ by the same
amount. This amount is known as the ________________ _________________ and is labeled with _____.
Directions: Determine whether each sequence is arithmetic or not. If it is, give the common difference, d.
a )  3, 2, 7,12,17,...
b)  4, 12, 24, 40,...
c) 1.9,1.2,.5, .2, .9,...
d ) 2, 4, 6,8,10,...
Note: if you graph the ordered pairs (n, an ) , of an arithmetic sequence, the points lie in a straight line, so you
can actually think of arithmetic sequences as _______________ functions. Let’s graph d) above.
Next, let’s discuss the difference between recursive arithmetic sequences and explicit arithmetic sequences.
Recursively, arithmetic sequences take the form _______________________.
Explicitly, they take the form an  a1  (n  1)d . What does (n – 1) represent?
1. Find the 20th term in the sequence 32, 25, 18, 11, …
2. Find the 11th term in the sequence 9.2, 9.15, 9.1, 9.05, …
Page 7 of 18
3. Find the missing terms in the arithmetic sequence 11, ____, ____, ____, - 17.
We need to find d: Step 1  Answer the 3 questions below
What is n?
What is an ?
What is a1 ?
Step 2  Use these to write an equation for an arithmetic sequence leaving d as the variable, then solve for
d and find the missing terms.
Another way to find d would be to take the total difference between the terms you have (11 and – 17 in this
case) and divide by the number of differences taken from a1 to an (which would be your n -1 value).
d
total difference

# of differences
4. Find the missing terms in the arithmetic sequence 17, ____, ____, ____, ____, - 13
5. Find the 11th term in an arithmetic sequence with a2  20.5 and a7  13
6. Find the 6th term in the arithmetic sequence with a9  120 and a14  195
Page 8 of 18
Day 4: Arithmetic Series
& Equation
An arithmetic series is the sum of the terms of an arithmetic sequence.
Find the value of the indicated series below.
1. S10 for 13  2  9  20  ...
15
2.
 (2k  5)
k 1
(10th partial sum)
3. Find the sum of the first 100 counting numbers.
Is there some way we can do these without having to know what each term’s value is and taking the time to
add them all up? Of course there is! Think about what happens with #3 above.
There are 3 things we need to know to find the value of an arithmetic series. They are—
1.
2.
3.
What we are really doing is finding the average of _____________________________
And multiplying by_______________________ in order to find the sum of an arithmetic series.
a a 
In other words, S n  n  1 n 
 2 
a) Can you now find S15 the sum for the series 25+12-1-14-…?
15
b) Find
 (14  3k )
k 1
Page 9 of 18
c) The center section of a concert hall has 15 seats in the first row and 2 additional seats in each
subsequent row.
How many seats are in the 20th row? (What are you finding here?)
How many seats are in the first 20 rows combined? (What are you finding here?)
d) Curtis has $50 and receives $8 a week for allowance. He wants to save all of his money to buy a new
mountain bike that costs $499 (including tax). Write an arithmetic sequence to represent the amount of
money he has after each week. Will Curtis have enough money to buy the bike after 1 year?
Page 10 of 18
Day 6: Geometric Sequences
& Equation
Question: What are the next 3 terms of the sequence below? What is the common difference, d?
8,12,16, 20, 24, 28,...
Today we will look at a new type of sequence: _______________________.
3, 6,12, 24, 48,...
What are the next 3 terms above? How is this sequence different from the one at the very top?
When successive terms of a sequence are found by multiplying by a given number (the number is called r for
_______________ ________________), then the sequence is a geometric sequence.
Type
Definition
Common ________
Example
Arithmetic
Geometric
Examples: Determine whether each sequence below is geometric, arithmetic or neither of those. If it is
geometric, find r, if it is arithmetic, find d.
a) 8,12,18, 27,...
b) 8,16, 24,32,...
d ) 1.7,1.3,.9,.5,...
e) 5,1,.2,.04,...
c) 6,10,15, 21,...
Two types of geometric sequences:
 Recursively, each term can be generated by an  an 1  r (multiply previous term by r)
 Explicitly, the nth term can be found by an  a1  r n 1
(multiply first term by r, n-1 times)
Page 11 of 18
1) Find the 9th term of the geometric sequence 5,10, 20, 40, 80,...
 Step 1: Find the common ratio 
 Step 2: Plug into the general equation  an  a1  r n 1
2) Find the 10th term of the geometric sequence
3 3 3
, , ,...
4 8 16
3) Find the 9th term of the geometric sequence with a5  96 and a7  384 .
 Step 1: Find the common ratio  a7  a5r  75
 Step 2: Find a1 using the general formula (don’t forget the  )
4) Find the 7th term of the geometric sequence with a4  8 and a5  40 .
Geometric Means are the terms between any 2 non-consecutive terms of a geometric sequence. If a and b
are positive terms of a geometric sequence with exactly 1 term between them, the geometric mean is equal to
ab .
Find the geometric mean between
1
1
and
.
2
32
Find the geometric mean between 16 and 25 .
Page 12 of 18
Extension: HW Start…
Day 7: Geometric Series – Finite and Infinite
& Equation
a) What’s the difference between a sequence and a series?
b) What is S 5 of 4  6  8 10 12 14 16 ?
5
c) Find:
 3n  1
n 1
Fortunately for us, there is a “short cut” for finding partial sums…
The nth partial sum (sum of the first n-terms) of a geometric series can be determined by the formula
 1 rn 
Sn  a1 
 where a1 is the first term and r is the common ratio.
 1 r 
Find the indicated sum for each geometric series below.
1. S12 for 3  6  12  24  ...
1
2.   
k 1  3 
5
k 1
Page 13 of 18
3. A six-year lease states that the annual rent for an office space is $84,000 for the first year and will
increase by 8% each additional year of the lease. What is the rent in the 6th year? What is the total rent
for the entire 6 year lease?
INFINITE GEOMETRIC SERIES
(ALSO KNOWN AS: GEOMETRIC SERIES WITH AN INFINITE NUMBER OF TERMS)
Directions: use the table below to find the first 6 partial sums of the following two series
1) S n 
1)
2)
1 1 1 1
   ...
2 4 8 16
n
an
1
2) S n 
2
1 1 1 1
   ...
32 16 8 4
3
4
5
6
Sn
an
Sn
Make a comparison of what would happen in these 2 cases if you were to continue to find these sums
infinitely:
A series __________________to a number S if the partial sums, S n get close to a real number as n gets
very large. In geometric series, this happens only if r  1 , that is “if r is between _______ and
________, a geometric series will __________________”.
If the sum of a series does not converge, we say it ________________. A geometric series will
___________________ if r  1 .
The sum of an infinite geometric series “S” with common ratio “r” and r  1 is given
a
by the formula S  1 .
1 r
Page 14 of 18
Determine whether each geometric series below converges or diverges.
1) 20  24  28.8  34.56  ...
1 1 1
 ...
2) 1   
3 9 27
3) 32 16  8  4  2...
If a series below converges, find its sum, S.
1) 5  4  3.2  2.56  ...

2)
k 1
1
3) 25  5  1   ...
5
2
3

k 1
2
4)   
k 1  5 
k
Page 15 of 18
Day 8: Review of Geometric – Quick Practice Quiz
& Equation
1. Write the recursive formula for finding an in a geometric sequence.
2. Write the explicit formula for finding an in a geometric sequence.
3. Write the formula for finding the nth Partial Sum of a geometric series.
4. Write the formula for finding the sum of a converging infinite geometric series.
5. When does a geometric series converge?
6. Tell what type of sequence: arithmetic, geometric or neither…
a. 5, 10, 20, 40, 80
b. 3, 5, 8, 12, 17, 23
c. 4, 6, 8, 10, 12
7. If a1  5 and r = - 2, find a6
8. Find the geometric mean between 25 and 4

9. Does the following series converge or diverge? Why?
1
  3   4 
n
n 1
Page 16 of 18
Day 9: Sequence and Series Test Review
& Equation
1.
In an arithmetic sequence, a2  6, a6  16 . Find a21 .
2.
In a geometric sequence, a3  9, a4  . Find
3.
In a sequence, a1  0, a2  4, a3  8 . Find a9 .
4.
Find the sum of the first 25 terms of the arithmetic series
11 + 14 + 17 + 20 + …
9
4
a7 .
1.__________________
2.__________________
3.__________________
4.__________________
5.
Find the sum of the first 10 terms of the geometric series
2 – 6 + 18 – 54 + …
5.__________________
6.
Does the series below converge? If it does, find the value it converges to
1 1
1
 

4 16 64
6.__________________
7
7.
Find the value of
 (4k  7)
k 1
7.__________________
Page 17 of 18
8.
Find x so that 25, x and 9 form a geometric sequence.
(in other words, find the geometric mean of 25 and 9)
8._________________
In the arithmetic sequence –8, -5, -2, … which term has a value of 52?
9.
9.__________________
10. Find the 30th term of the arithmetic sequence whose 17th term is 7 and 47th term is 31.
10.____________
11.Find the value of the 2 series below:
1
a.  2  
k 1  3 
6
k 1
________________________

1
b.  2  
k 1  3 
k 1
______________________
12.A display of cans in the supermarket has 20 cans on the bottom row, 18 cans on the
next row, 16 on the next, and so on. If there are 7 rows of cans in all, how many cans
are there in the stack? Give your answer by writing the sequence/series used in the
problem.
Page 18 of 18