Sequences & Series
Teacher:___________________________________________Pd.:_______________
Table of Contents
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Arithmetic Sequences
Homework
1-7
Arithmetic Series
Homework
10 - 14
Summation & Sigma Notation
Homework
18 - 22
Geometric Sequences
Homework
24 - 28
Geometric Series
Homework
30 - 36
Review
39 – 44
PROJECT – Create Your Own Reality Series
45 - 46
8-9
15 - 17
23
29
37 – 38
Algebra 2/Trig. Apps
Arithmetic Sequences & The Explicit Formula (Day 1)
SWBAT: Find the nth term of an arithmetic sequence
Find the position of a given term in an arithmetic sequence
Find the arithmetic means
WARM – UP
Find the next four terms of each arithmetic sequence
a. 26, 21, 16, …
b. 34, 40, 46, . . .
c. 62, 57, 52, . . .
Definition
An arithmetic sequence is a sequence in which each term after the first is found by adding a
constant, called the _______________________________ ______________________________ ____, to the
previous term.
Sequence
Expressed
in Terms
of d and
the First
Term
91
83
75
67
...
symbols
a1
a2
a3
a4
...
an
numerical
91+0(-8)
91 + 1(-8)
91 + 2(-8)
91 + 3(-8)
...
an = a1 + (n – 1) d
symbols
a1 + 0  d
a1 + 1 d
a1 + 2  d
a1 + 3  d
...
an = a1 + (n – 1)  d
numerical
Explicit Formula for the nth
Term of an Arithmetic
Sequence
The nth term an of an arithmetic sequence with first term a1
and common difference d is given by
an = a1 + (n – 1)d
where d is a positive integer
1
Example 1 Find the nth Term Using an Explicit Formula
In radio broadcasting, autumn is one of the most critical periods for ratings. Radio stations
try to pull in as many listeners as they can by using a variety of gimmicks. In the fall of
2001, a radio station had a contest in which listeners had a chance to win $1000 every
hour. In order to win, listeners needed to call in and correctly answer a contest question.
The contest started with $1000 and $97 was added for the next caller each time the
previous caller answered the question incorrectly. Suppose you were the 18th caller and
the fist to answer the question correctly. How much money would you win?
a1 = ______________ and d = __________. a18 represents the cash the 18th caller will win. So,
n = __________
Find a18 using an = a1 + (n – 1) d.
If you were the 18th caller and you answered the question correctly, you would win
$__________________
You Try:
Suppose a race car driver increases her speed at a constant rate. What will her speed be
after 20 seconds if her initial speed is 65 mph and her rate of acceleration is 4.8 mph?
a1 = ______________
d = _______________
n = _______________
2
Sometimes you may know two terms of a sequence, but they are not consecutive terms of
the sequence. The terms between any two nonconsecutive terms of an arithmetic sequence
are called arithmetic means. In the sequence below, 32, 41, and 50 are the three
arithmetic means between 23 and 59.
Example 2 Finding the Arithmetic Means
a. Find the four arithmetic means between 18 and 78.
b. Graph the sequence using the x-axis for the number of the term and the y-axis for the
term itself.
a. Use the nth term formula to find the common difference. In the sequence
18, _?__, __?__, __?__, __?__, 78,
18 is a1 and 78 is a____
an = a1 +(n – 1)d
Now use the value of d to find the four arithmetic means.
The arithmetic means are ______________________________________________________________
Term
Number of Term
3
You Try
a. Find the six arithmetic means between 12 and 47.
b. Find the missing terms of the sequence _________, 17, ______, ______, ______, -7.
Example 3 Writing an equation for the nth Term
Write an equation for the nth term of the arithmetic sequence 6, 13, 20, 27, . . .
a1 = _______ d = _______. Use the nth term formula to write the equation.
an = a1 + (n – 1)d
an = __________________________
The equation is an = _______________________
The graph of the line given by this equation
contains the terms of the arithmetic sequence.
Compare the slope of the line described by this equation and the value of the common
difference.
4
Check For Understanding
1. Explain why the sequence 4, 5, 7, 10, 14, . . . is not arithmetic.
2. Explain whether 24 is a term in the sequence represented by an = 5 + (n – 1)2.
3. Refer to the graph of the arithmetic sequence at the right.
a. Write the first five terms of the sequence.
b. Explain why the points are not connected.
c. What is the equation of the line that passes
through these points?
d. State the slope of the line in part c and explain how
it relates to the nature of the sequence.
4. Andrea says that 220 is a term of the sequence an = 16 + (n-1)12. Janice says that this is
impossible. Who is correct?
Explain.
Guided Practice
Find the first five terms of each arithmetic sequence described.
5. a1 = 5, d = 3
6. a1 = 14, d = -2
Find the nth term of each arithmetic sequence.
7. a1 = 3, d = -5, n = 24
8. a1 = -5, d = 7, n = 13
9. Complete: 68 is the _______ th term of -2, 3, 8, . . .
5
10. Find a13 for the arithmetic sequence -17, -12, -7, . . .
11. a. Find the three arithmetic means between 44 and 92.
b. Graph the sequence using the x-axis for the number of the term and the y-axis for the
term itself.
12. Write an equation for the nth term of the arithmetic sequence -26, -15, -4, 7, . . .
SUMMARY
Rather than write a recursive formula for an arithmetic sequence, we can write an explicit
formula. To write the explicit we use:
an = a1 + (n-1)d
• an is the nth term of the sequence. When writing the general expression for an arithmetic
sequence, you will not actually find a value for this. It will be part of your formula
much in the same way x’s and y’s are part of algebraic equations.
• a1 is the first term in the sequence. To find the explicit formula, you will need to be given
(or use computations to find out) the first term and use that value in the formula.
• n is treated like the variable in a sequence. For example, when writing the general
explicit formula, n is the variable and does not take on a value. But if you want to
find the 12th term, then n does take on a value and it would be 12.
• d is the common difference for the arithmetic sequence. You will either be given this
value or be given enough information to compute it. You must substitute a value for
d into the formula. You must also simplify your formula as much as possible.
6
EXIT TICKET
What is a formula for the nth term of sequence B shown below?
B = 10, 12, 14, 16, . . .
1) bn = 8 + 2n
2) bn = 10 + 2n
3) bn = 10(2)n
4) bn = 10(2)n - 1
7
Homework Day 1
Find the nth term of each arithmetic sequence.
1. a1 = 3, d = 7, n = 14
2. a1 = -4, d = -9, n = 20
1
5
3. a1 = 5, d = 3, n = 12
4. a1 = 2,
3
d = − 2 , 𝑛 = 11
Complete each statement
5. 170 is the ___?____ th term of -4, 2, 8, . . .
6. 124 is the ___?___th term of -2, 5, 12, . . .
7. 166 is the ___? ___th term of 30, 34, 38, . . .
Find the indicated term in each arithmetic sequence.
8. a12 for – 17, -13, -9, . . .
9. a21 for 121, 118, 115, . . .
10. a12 for 8, 3, -2, . . .
Find the arithmetic means in each sequence. Then graph each sequence using the x-axis for
the number of the term and the y-axis for the term itself.
11. 10, __?__, __?__, - 8
12. __?__, -5, __?__. __?__, 4, __?__
8
Write an equation for the nth term of each arithmetic sequence.
13. 7, 16, 25, 34, . . .
14. 18, 11, 4, -3, . . .
15. -3, -5, -7, -9, . . .
16. a. The first term of an arithmetic sequence 7, and each term is 4 more than the
previous term. Find the value of the eighth term.
b. Write a formula for the nth term of this sequence.
STANDARDIZED TEST PRACTICE
17. What number follows 20, in this arithmetic sequence?
8, 11, 14, 17, 20, . . .
A. 5
B. 23
C. 26
D. 29
18. Find the first term in the arithmetic sequence.
1
2
1
___________, 8 3, 7, 5 3, 4 3
A. 3
2
B. 9 3
1
C. 10 3
D. 11
19. Geologists have calculated that the continents of Europe and America are drifting apart at an
average of 12 miles every 1 million years. That’s an average of 0.75 inch a year. If the
continents continue to drift apart at the same rate, how many inches will they drift in 50
years? (Hint: a1 = 0.75)
20. People used to believe that the heavier an object was, the faster it would fall. Galileo proved
that this was incorrect. He dropped two different weights simultaneously from the Leaning
Tower of Pisa, and they both hit the ground at the same instant. When an object dropped from
a tall building, no matter how much it weighs, it falls 16 feet in the first second, 48 feet in the
second second , and 80 feet in the third second. How many feet would a falling object fall in
the tenth second?
9
Algebra 2/Trig. Apps
Arithmetic Series (Day 2)
SWBAT: Find sums of arithmetic series
Find specific terms in an arithmetic series
Warm-Up
Find the nth term of each arithmetic sequence.
1. a1 = 7, d = 3, n = 14
2. a1 = 20, d = 4, n = 100
3. Find the 12th term of the sequence -17, -13, -9, . . .
4. Which term of the sequence -2, 5, -12, . . . is 124?
5. Find the arithmetic means of the sequence 55, ___, ___, ___, 115.
Suppose I place pennies in this empty jar according to an
arithmetic sequence such as 1 the first day, 2, the next day, 3
the third day, 4 on the fourth day …
Is there a way, other than counting the pennies to determine
the total number added after a certain number of days?
The sum of the terms of a sequence is called a series. The arithmetic series representing the
sum of the pennies above would be
1+2+3+4+5+...
10
Arithmetic Sequence
Arithmetic Series
4, 7, 10, 13, 16
-10, -4, 2
4 + 7 + 10 + 13 + 16
-10 + (-4) + 2
2 6 10 14
, , ,
7 7 7 7
2 6 10 14
+ + +
7 7 7 7
The symbol Sn is used to represent the sum of the first n terms of a series. For example, S3 means
the sum of the fist three terms of a series. In the series 4 + 7 + 10 + 13 + 16, S3 would be 4 + 7 + 10
or 21.
If a series has a large number of terms, it is not convenient to list all the terms and then find their
sum. To develop a general formula for the sum of any arithmetic series, let’s consider the series of
S7 = 1 + 2 + 3 + 4 + 5 + 6 + 7
Suppose we write S7 in two different orders and find the sum.
S7 = 1 + 2 + 3 + 4 + 5 + 6 + 7
+ S7 = 7 + 6 + 5 + 4 + 3 + 2 + 1
2 S7 = 8 + 8 + 8 + 8 + 8 + 8 + 8
7 sums of 8
2 S7 = 7(8)
7
S7 = 2 (8) Divide each side by 2
7
Now let’s analyze what these numbers represent in terms of Sn. In the equation S7 = 2 (8), 7
represents n and 8 represents the sum of the first and last terms, a1 and an. Thus, we can replace
𝑛
the equation with the formula Sn = 2 (a1 + an). This formula can be used to find the sum of any
arithmetic series.
The sum Sn of the first n terms of an arithmetic series is given by
𝒏
Sn = 𝟐 (a1 + an), where n is a positive integer.
11
Example 1
Find the sum of the first 50 positive even integers.
𝑛
Sn = 2 (a1 + an).
a1 = 2, n = 50, and an = 100
The sum of the first 50 positive even integers is _________________________
Recall: an = a1 + (n – 1)d. Using this formula and substitution gives us another version of
the formula for the sum of an arithmetic sequence.
𝑛
Sn = 2 (a1 + an)
𝑛
= 2 {a1 +[a1 + (n – 1)d]}
𝑛
= 2[2a1 + (n – 1)d]
You can use this formula when you do not know the value of the last term.
Example 2
Find the sum of the first 40 terms of an arithmetic series in which a1 = 70 and d = -21
12
Example 3
Refer to Homework problem 20 from the previous lesson. A
free-falling object falls 16 feet in the first second, 48 feet in
the second second, 80 feet in the third second, and so on.
How many feet would a free-falling object fall in 20 seconds
if air resistance is ignored?
To find the total distance fallen by an object, add the first 20
terms of the sequence 16, 48, 80, . . .
CHECK FOR UNDERSTANDING
1. Explain when it is necessary to use both versions of the sum formula to solve a
problem. ____________________________________________________________________________
_______________________________________________________________________________________________________
Practice:
State the first term, the common difference, the last term, and the number of terms for each
arithmetic series.
1. 7 + 1 + (-5) + (-11)
2. a1 + 35, + 38, + 41 + 44
Find Sn for each arithmetic series described.
3. a1 = 4, an = 100, n = 25
4. a1 = 132, d = -4, an = 52
13
4. Find the sum of each described series.
a. first 60 terms in which a1 = 15 and d = 80
b. first 15 odd integers
5. A supermarket display consists of six rows of stacked boxes. Each row has three fewer boxes
than the one below it. The top row has 35 boxes. How many boxes are in the display?
SUMMARY
There are two ways to find the sum of a finite arithmetic sequence. To use the first
method, you must know the value of the first term a1 and the value of the last term an.
𝒏
Then, the sum of the first n terms of the arithmetic sequence is Sn = 𝟐 (a1 + an). To use the
second method, you must know the value of the first term a1 and the common difference
𝑛
d. Then, the sum of the first n terms of an arithmetic sequence is 2[2a1 + (n – 1)d].
EXIT TICKET
18 + 22 + 26 + 30 + . . . + 50 = ?
A. 146
B. 272
C. 306
D. 340
14
Algebra 2/Trig. Apps Homework Day 2
State the first term, the common difference, the last term, and the number of terms for each
arithmetic series.
1. 6 + 7.4 + 8.8 + 10.2 + 11.6
2. a1 + 12 -6 – 24 -42 -60 +a7
Find Sn for each arithmetic series described.
3. a1 = 40, n = 20, d = -3
4. d = 5, n = 16, an = 72
5. a1 = 58, an = -7, n =26
6. a1 = 3, an = -38, n = 8
1
7. a1 = 12, d = 3, n = 13
1
9. a1 = -2, d = 3, an = 9
8. d = 7, n = 18, an = 72
1
10. a1 = 5, d = 2, n = 13
15
Find the sum of each arithmetic series.
11. 5 + 11 + 17 + . . . + 95
12. 38 + 35 + 32 + . . . + 2
13. Find the sum of each described series.
a. first 60 terms in which a1 = 15 and d = 80
b. first 15 odd integers
14. A company responsible for laying the foundation of a new high-rise office building must pay
$5000 a day for the first five days that the completion of the foundation is late. On the sixth
day and each day thereafter, the penalty is increased by $200 a day. If the company was
penalized $65,600 how many days late were they in laying the foundation?
15. Some theatres have seats with limited viewing. Suppose only the people in the first 24 rows
of a certain theatre have 100% visibility. In the this section, there are 20 seats in the first
row, and each subsequent row has one more seat than the row in front of it. How many seats
are there in this theatre where people can see the whole show?
16
Review
16. The U.S. Department of Agriculture recommends two to four servings of fruit daily. Fruit is
a good source of fibber as well as vitamins and minerals. The chart at the right displays
the fiber content for one serving size of each fruit listed.
a. Find the range of the data.
b. Find the quartiles of the data.
c. Find the interquartile range of the data.
d. Name any outliers in the data.
e. Make a box-and-whisker plot of the data.
17
Algebra 2/Trig. Apps
Summation & Sigma Notation for Arithmetic Series – Day 3
SWBAT: Use sigma notation to express sums
Find the sum Sn for each arithmetic series described.
1. a1 = 4, an = -16, and n = 5.
2. a1 = 3, an = 33, and n = 6.
3. Find the sum of the series -4 + (-1) + 2 + 5 + . . . + 53.
Writing out a series is often time-consuming and lengthy. To simplify this, mathematicians use a
more concise notation called sigma or summation notation.
2 + 4 + 6 + 8 + . . . + 20 can be expressed as
10
å 2n
n=1
This expression is read the sum of 2n as n increase from 1 to 10.
When using sigma notation, the variable defined below the
å
(sigma) is called the index of
summation. The upper number is the upper limit of the index. To generate the terms of the
series, successively replace the index of summation with consecutive integers as values of n. In
this series, the values of n are 1, 2, 3, and so on, through 10.
18
Example 1 Write the terms of
7
å (2k + 5) and find the sum.
k=3
Method 1
Replace k with 3 and then with 4, 5, 6,
and 7 to find the terms
7
å (2k + 5) = [2(3) + 5] + [2(4) + 5] +
k=3
[2(5) + 5] + [2(6) + 5] +
[2(7) + 5]
Method 2
𝑛
Use Sn = (𝑎1 + 𝑎𝑛 ).
2
n = 5, a1 = 11
an = a5 = 2(7) + 5 or 19
5
Sn = (11 + 19)
2
= 75
= 11 + 13 + 15 + 17 + 19
= 75
The sum of the series is 75.
Practice:
Write the terms of each series and find the sum.
a.
6
å (3n + 7)
n=1
b.
4
å (3i - 6)
i=1
19
You can use the sum and sequence functions on a graphing calculator to find the
sum of an arithmetic series.
20
Practice:
Use a graphing calculator to find the sum of each series.
a.
20
å (4 + 3n)
b.
n=1
9
å (5k + 4)
c.
k=3
35
å (4t -17)
t=2
Just as a polynomial can be expressed in more than one form, the summation of a
series can be expressed in different ways. The summation of the series in Example 1
is expressed as
7
å (2k + 5) .
It can also be expressed as
k=3
7
10
k=3
k=6
å[7 + 2(k -1)] or å (2k -1).
Explain why?
________________________________________________________________________________________
CHECK FOR UNDERSTANDING
1. Define the term indicated sum. ________________________________________________________
______________________________________________________________________________________________
2. Explain the purpose of the index of summation. _____________________________________
______________________________________________________________________________________________
3. Write the terms of
7
å 2n and find the sum.
n=1
21
SUMMARY
EXIT TICKET
22
Algebra 2/Trig Homework Day 3
Write the terms of each arithmetic series and find the sum.
1.
6
å (2n +11)
2.
n=1
23
å (5t - 3)
3.
t=19
11
å (42 - 9k)
k=7
Use a graphing calculator to find the sum of each arithmetic series.
4.
75
å (2n + 5)
50
å (3n -1)
5.
n=21
6. a. Evaluate
n=10
6
å (a - 2)2 and
a=3
4
åa
2
.
a=1
b. What do you notice?
c. Why does this work?
7.
20
å (2n +1)
n=1
9.
4
å (2k - 3)
k=1
8.
25
å x -1
n=5
10.
5
å (k
2
+ 2)
k=1
23
Algebra 2/Trig. Apps
Geometric Sequences – Day 4
SWBAT: To find the nth term of a geometric sequence using an explicit formula
To find the position of a given term in a geometric sequence
Warm – Up
1. Find the sum of the series
a.
5
å (5k - 7)
k=0
b.
1
å (4 p -1)
p=-2
2. Use sigma notation to represent the given series.
a. 2 + 4 + 6 + . . . for 45 terms.
b. 8 + 13 + 18 + 23 + . . . for 10 terms
3. Write a recursive formula for the given sequence and find the next two terms.
a. 4, 16, 64, . . .
b. 81, 27, 9, . . .
4. Find the nth term of the sequence.
a1 = -2, n = 5, and r = 3
5. A certain model automobile depreciates 20% of its value each year. If it costs $16,800 new,
what is its value at the end of eight years?
24
If you have ever bounced a ball, you know that
when you drop it, it never rebounds to the height
from which you dropped it. Suppose a ball is
dropped from a height of three feet, and each time
it falls, it rebounds to 60% of the height from
which it fell. The heights of the ball’s rebounds
form a sequence.
Why is this sequence not an arithmetic sequence?
Compare a common difference and a common ratio.
_________________________________________________________________________________________________
The height of the first rebound of the ball is 3(0.6) or 1.8 feet. The height of the
second rebound is 1.8(0.6) or 1.08 feet. The height of the third rebound is 1.08(0.6)
or 0.648 feet. The sequence of heights, 1.8, 1.08, 0.648, …, is an example of a
geometric sequence. Recall, a geometric sequence is a sequence in which each
term after the first is found by multiplying the previous term by a constant r called
the common ratio.
As with an arithmetic sequence, you can label the terms of a geometric sequence as
a1, a2, a3 and so on. The nth term is an and the previous term is an – 1.
Explicit Formula for the nth term of a Geometric
Sequence
The nth term an of a geometric sequence with first term a1 and common
ratio r is given by the explicit formula
an = a1  r n – 1
where n is any positive integer.
25
Example 1 Find a Particular Term
Find the eighth term of a geometric sequence for which a1 = -3 and r = -2
an = a1  rn – 1
a8 = (
a8 = (
)  (
) (
Formula for nth term
)(
)
n = _______, a1 = __________, r = _____________
)
a8 = __________
The eighth term is ________________
Example 2 Write an Equation for the nth Term
Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, . . .
a1 = _________ r = _________
Example 3 Find a Term Given the Fourth Term and the Ratio
Find the tenth term of a geometric sequence for which a4 = 108 and r = 3.
First, find the value of a1.
Now find a10
26
PRACTICE:
1. Find the sixth term of a geometric sequence for which a1 = -3 and r = -2.
2. Write an equation for the nth term of the geometric sequence 5, 10, 20, 40, . . .
3. Find the seventh term of a geometric sequence for which a3 = 96 and r = 2.
CHECK FOR UNDERSTANDING
1. Find the Error Marika and Lori are finding the seventh term of the geometric
sequence 9, 3, 1, . . .
Who is correct? Explain your reasoning.
2. Explain how you know if a list of numbers forms a geometric sequence.
27
SMMARY
A geometric sequence is a sequence in which each term after the first can be found
by multiplying the preceding term by a constant, called the common ratio. The
common ratio, r, can be found by diving any term by the preceding term.
nth Term of the Geometric Sequence
The nth term, an, of a geometric sequence with first term
a1 and common ratio r is given by either of the following
formulas.
an = an – 1 r or an = a1 rn – 1
EXIT TICKET
1
Find the 4th term of the geometric sequence for which a1 = -8 and r = - .
2
[A] −
1
2
[B] −
21
2
[C] 1
[D] -10
28
Algebra 2/Trig Apps. Homework Day 4
Find the indicated term of each geometric sequence
1. a9 for the geometric sequence 60, 30, 15, . . . .
2. a1 = 7, r = 2, n = 4
3. a3 = 32, r = 0.5, n = 6
4. Write an equation for the nth term of the geometric sequence 4, 8, 16, . . .
Find the nth term of each geometric sequence.
1
1
5. a1 = 3, n = 8, r = 3
6. a1 = 64, n = 9, r = 4
1
7. a1 = 4096, n = 8, r = 4
9. a6 = 3, n = 12, r = 2
8. a4 = 16,, n = 8, r = 0.5
1
1 1
10. a7 for 32, 16, 8, . . .
Find the value(s) of y that makes each sequence geometric.
(Hint: determine r first)
11. 2, 8, 32, 5y + 3, . . .
12. 3, 6, 2y + 18, . . .
13. Iodine-131 is used medically to study the activity of the thyroid gland. Iodine – 131 has a
half-life of about 8 days. This means that approximately every 8 days, half the mass of the
iodine decays into another element. If a container held a mass of 64 milligrams of iodine – 131,
how much is left after 40 days?
29
Algebra 2/Trig Apps. Geometric Series Day 5
SWBAT: Find sums of geometric series
Find specific terms in a geometric series
WARM – UP
Find the nth terms of each described geometric sequence
1. a1 = -2, n = 5, and r = 3
2. a1 = -5, n = 4, and r = 2
How is e-mailing a joke like a geometric series?
How many people have read your joke at the end of Monday?
at the end of Tuesday?
at the end of Wednesday?
at the end of Thursday?
30
Notice that every day, the number of people who read your joke is three times the
number that read it the day before. By Sunday, the number of people, including
yourself, who have read the joke is 1 + 3 + 9 + 27 + 81 + 243 + 729 + 2187 or 3280!
The numbers 1, 3, 9, 27,81, 243, 729, 2187 form a geometric sequence in which
a1 = 1 and r = 3. Since 1, 3, 9, 27,81, 243, 729, 2187 is a geometric sequence, 1 + 3 +
9 + 27 + 81 + 243 + 729 + 2187 is called a geometric series. Below are some more
examples of geometric sequences and their corresponding geometric series.
Geometric Sequence
1, 2, 4, 8, 16
4, - 12, 36
1 1
5, 1, ,
5 25
Geometric Series
1+ 2 + 4 + 8 + 16
4 + (-12) + 36
1
1
5+1+ +
5
25
To develop a formula for the sum of a geometric series, consider the series given in
the e-mail situation above.
𝑎1− 𝑎 𝑟 8
1
The expression for S8 can be written as S8 =
. A rational expression like this
1−𝑟
can be used to find the sum of any geometric series.
Remember a9 can also be written as a1 r 8
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You cannot use the formula for the sum with a geometric series for which r = 1
because division by 0 would result. In a geometric series with r = 1, the terms are
constant. For example, 4 + 4 + 4 + . . . + 4 is such a series. In general, the sum of n
terms of a geometric series with r = 1 is n a1.
Example 1 Find the sum of the first six terms of the geometric series for which
a1 = 5 and r = -2.
As with arithmetic series, you can use sigma notation to represent geometric series.
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Example 2 As a hot-air balloon rises from Earth’s
surface, the air in the balloon cools. If
the air is reheated, the balloon rises
more slowly with each minute of flight.
Suppose that after 1 minute, a hot-air
balloon rises 120 feet. In each
succeeding minute, the balloon rises
only 60% as far as it rose in the previous
minute. How far will the balloon rise in
8 minutes?
Example 3 Evaluate a Sum Written in Sigma Notation
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PRACTICE:
1. Find the sum of each described geometric series.
1
1
a. a1 = 16, r = , and n = 7
b. a1 = 4, r = − and n = 8
2
2
2. Using the information in example 2 - How far will the balloon rise in 10 minutes?
3. Write the terms of
8
1 2s-9
(3)
å
s=5 2
8
1
å 2 (3)
2s-9
s=5
How can you find the sum of a geometric series if you know the first and last terms
and the common ratio, but not the number of terms? Remember the formula for the
nth term of a geometric sequence or series, an = a1  rn – 1. You can use this formula
to find an expression involving rn.
Now substitute an r for a rn in the formula for the sum of a geometric
series.
The result is
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Example 4 Use the Alternate Formula for a Sum
1
Find the sum of a geometric series for which a1 =729, an = -3 and r = − .
3
Since we do not know the value of n, use the alternate formula
Example 5 Find a1 in a geometric series for which S7 = 70,993 and r = 4
Sn =
𝑎1(1− 𝑟𝑛 )
1−𝑟
You Try:
4. Find the sum of a geometric series for which a1 = 4, an = 256, and r = 4.
5. For a certain geometric series, S8 = 13, 120 and r = 3. Find a1 for the series.
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SUMMARY
The indicated sum of the terms of a geometric sequence is called a geometric
series.
Sum of a Geometric Series
The sum Sn of the first n terms of a geometric series is given by
Sn =
𝑎1− 𝑎1 𝑟 𝑛
1−𝑟
or
𝑎1 (1−𝑟 𝑛 )
Sn =
1−𝑟
, where r ≠ 1
Exit Ticket
Find Sn for the geometric series described.
a1 = 12, a5 = 972, r = -3
36
ALGEBRA 2/TRIG. APPS HOMEWORK DAY 5
Find Sn for each geometric series described.
1
1. a1 = 81, r = 3, n = 6
2
4. a1 = 243. r = − 3, n = 5
2. a1 = 1, r = -2, n = 7
3. a1 = 5, r = 4, n = 5
5. a1 = 8, r = -2, n = 7
6. a1 = 2, r = -3, a5 = 162
Find the sum of each geometric series.
7. 54 = 36 + 24 + 16 + . . . to 6 terms
8.
3 – 6 + 12 - . . . to 7 terms
9. 12 + 12 + 12 + . . . to 12 terms
10. 2401 – 343 + 49 - . . . to 5 terms
Write the terms of each geometric series and find the sum.
5
1
11. å × 2 n-1
n=1 4
æ 1ö
12. Write the terms of å 81ç ÷
è 3ø
n=1
7
n-1
and find the sum.
Find a1 for each geometric series described
215
1
13. Sn = -364, r = -3, n = 6
14. Sn =
, r = − 2, n = 7
64
15. Sn = 315, an = 5, r = 0.5
16. Sn = 165, an = 48, r = − 3
2
37
Express each series in sigma notation and find the sum.
17. 75 + 15 + 3 + . . . to 10 terms
18. 243 + 162 + 108 + 72 + to 12 terms
19. Raheem is helping his father install a fence. He uses a sledgehammer to drive pointed fence
posts into the ground. On his first stroke, he drives the post 4 inches into the ground. The soil
is denser the deeper he drives, so each stroke after the first, he can only drive the post 40% of
the distance he did on the previous swing of the hammer. After 10 strokes, hw far has he
driven the post into the ground?
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ALGEBRA 2/TRIG APPS REVIEW DAY 6
1. Find the 11th term of 5, 8, 11, . . .
1. Find the sum of the first 8 positive
odd integers.
2. Evaluate: å (2 k -1)
2. Find the sum of 100 terms of the
geometric sequence
1, -1, 1, -1, 1, -1, . . .
3. Which term of this arithmetic
sequence is 177?
3. Which term of this arithmetic
sequence is 25?
-2, 1, 4, . . .
5
k=1
2, 9, 16, . . .
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4. Find the sum of the first 10 positive
integers.
4. Find the 9th term of the sequence
3, 7, 11, . . .
5. Find the 7th term of an arithmetic
sequence whose first term is -8 and
whose common difference is 3.
5. Evaluate:
6. Evaluate:
6. Which term of this sequence is 275?
5, 10, 15, . . .
10
å ((2k +1)
0
-1)
4
å (3k -1)
k=1
k=2
7. Find the 7th term of the sequence
1, 2, 4, . . .
7. Evaluate:
4
å2
n
n=1
8. For the sequence 2, 4, 8, 16, . . . the
value of S4 is __________________
8. Find the 10th term of the sequence
-6, 1, 8, . . .
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ALGEBRA 2/TRIG. APPS REVIEW
Find the nth term of each arithmetic sequence.
1. a1 = 6, d = 8, n = 5
4. a1 = -2, d = -3, n = 15
2. a1 = -5, d = 7, n = 22
5. a1 = 4, d = 3, n = 32
3. a1 = 5, d = -2, n = 9
6. a1 = 8, d = -5, n = 10
Complete each statement
7. 72 is the ___?__ th term of -5, 2, 9, . . .
9. 49 is the __?__ th term of 4, 9, 14, …
Find the arithmetic means
11. – 7, __?__, __?__,__?__, 9
13. __?__, 6, __?__, __?__, -3, __?__
8. – 37 is the __?__th term of 1, -1, -3, -5, …
10. −
17
4
1
3
is the __?__th term of 24, 2, 14, . . .
12. 12, __?__, __?__, 4
14. __?__, 49, __?__, __?__, 28
41
Find Sn for each arithmetic series
15. a1 = 12, an = 117, n = 36
16. 4 + 10 + 16 + …+ 106
17. a1 = 85, an = 25, n = 21
18. 10 + 4 + (-2) + . . . + (-50)
19. Write the terms of the arithmetic series
20. Find the first three terms of an arithmetic
series if a1 = 3, an = 24, and Sn = 108.
13
13
n=2
n=2
å (3n +1) å (3n +1) and find the sum.
42
Find the indicated term in each geometric sequence
2 4 8
22. an for a1 = 2, r = 2, and n = 5
21. a6 for 3 , 3 , 3,…
23. an for a1 = 7, r = 2. and n = 4
1
24. an for a1 = 243, r = − 3 and n = 5
Find Sn for each geometric series described
25. a1 = 12, r = 3, n = 5
26. a1 = 4, r = − 2, n = 6
27. a1 = 256, r = 0.75, n = 9
28. a1 = 1, a5 = 16, r = − 2
1
1
1
3
29. a1 = 625. a5 = 81, r = 5
Find specific terms in a series
Concept Summary
Find a1 if S5 = 2.75 and r = 2
2.75 =
𝑎1 (1−(−2)5 )
1−(−2)
11 𝑎(33)
=
4
3
33 = 132a1
1
= 𝑎1
4
29
43
Find a1 for each geometric series described
29. Sn = 1-31. r = 5, n = 5
30. Sn = 30, r = -2, n = 4
31. Sn = -61, r = -1, n = 5
32. Sn = 244, r = -3, n = 5
2
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Create Your Own Reality Series!
Sequences and Series – Project
Due Date: __________________________
Would you like to create your own reality series? What
would you do?
What kinds of people or things might be involved? Maybe
you could focus on the number of seats in a theatre or the
distance a ball travels when it bounces. What’s that? You
thought this project was going to be about a television reality
series? Sorry, this project focuses on arithmetic and geometric
series (and sequences). In some cases, this might be more
interesting than what you find on television!
Arithmetic and geometric sequences and series arise in many situations.

Depreciation of a car can be an arithmetic sequence if the car depreciates by a certain dollar
amount every year. The sequence becomes geometric if the car depreciates by a percentage each
year.

Investments can be represented by arithmetic sequences or series if a set dollar amount is added at
given intervals.

If salary increases are given at a particular percentage per year, a geometric sequence can be used.

It is also interesting to calculate the distance traveled by a ball as it bounces. If a ball bounces to
80% of its previous height, you can use this as the common ratio to evaluate the geometric series.
Be sure to take into account the distance up and down between each bounce.

An athlete in training might add a set distance to each workout. An arithmetic series can be used to
calculate the total distance after one month of training.
In our studies of Sequences and Series, we discussed many topics about arithmetic and geometric
sequences and series. You have seen how to find the nth term, write explicit and recursive formulas and
find the sum of each type of series.
Your task is to choose situations that can be represented by arithmetic or geometric sequences and series.
Choose one situation that is arithmetic and one that is geometric. If you wish, search the Internet for reallife scenarios to help you get started. If you choose to use something you find during your research, be sure
to change the numbers to make your application unique. Find something fun that yu find interesting.
Whether it is investment related, physics related, or something else entirely, choose what interests you
most to make your own reality series!
The creation of a well-developed Reality Series Project should have:
1. Real-life series: Choose real-life situations which use arithmetic or geometric sequences and
series. You must have at least one arithmetic and geometric for this project. Write a
45
description of each scenario.
2. Research: Cite any research you did to discover the real-life applications of sequences and
series. If you created the real-life applications yourself, explain your thinking. If you used
ideas from other sources, show how you changed the terms, common difference, or common
ratio to make your application unique.
3. Diagrams or pictures: Include a diagram or picture of the situations you have chosen. Write
out the 1st several terms and use pictures to represent what is taking place. For example, if a
ball is bouncing you might want to show the distance traveled in the 1st several bounces.
4. Formulas: Write the recursive and explicit formulas for each sequence in the series. Then
write the series using summation notation.
5. Presentation: Create a visual to display your series. Suggestions of visuals include but are not
limited to: create a poster, a pamphlet, a power point presentation, or a website. If you create
an electronic visual a copy must be printed and turned in to be graded.
Your project will be graded using the following grading rubric:
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