Inferences and Conclusions from Data

1. OPEN ENDED Write an arithmetic series for which S5 = 10.
2. OPEN ENDED Write a geometric series for which r = ½ and n = 4.
3. Writing in Math Use the information on page 649 to explain how arithmetic series apply to amphitheaters. Explain what the sequence and the series that can be formed from the given numbers represent, and show two ways to find the seating capacity of the amphitheater if it has ten rows of seats.
4. Writing in Math Use the information on page 663 to explain how e-mailing a joke is related to a geometric series. Include an explanation of how the situation could be changed to make it better to use a formula than to add terms.

Q
©
Arithmetic Series
GET READY for the Lesson
Austin, Texas has a strong musical
tradition. It is home to many indoor and
outdoor music venues where new and
established musicians perform regularly.
Some of these venues are amphitheaters
that generally get wider as the distance
from the stage increases.
on.
ley
ary
La
ਸਲਾ
on
Suppose a section of an amphitheater can
seat 18 people in the first row and each
row can seat 4 more people than the
previous row.
din
Arithmetic Series The numbers of seats in the rows of the amphitheater
form an arithmetic sequence. To find the number of people who could sit
in the first four rows, add the first four terms of the sequence. That sum
is 18 + 22 + 26 + 30 or 96. A series is an indicated sum of the terms of a
sequence. Since 18, 22, 26, 30 is an arithmetic sequence, 18 + 22 + 26 + 30
is an arithmetic series.
S, represents the sum of the first n terms of a series. For example, S4 is the
sum of the first four terms.
Tip
d Sum
a series is
vhen the
le series are
indicated
expression
ates the
ich includes
+ or –
To develop a formula for the sum of any arithmetic series, consider the
series below.
S, = 4 + 11 + 18 + 25 + 32 +39 +46 +53 + 60
Write S, in two different orders and add the two equations.
Sy = 4 + 11 + 18 + 25 + 32 +39 +46 +53 + 60
(+) S, = 60 + 53 + 46 +39 + 32 + 25 + 18 + 11 + 4
28, = 64 + 64 + 64 + 64 + 64 + 64 + 64 + 64 + 64
GET READY for the Lesson
Suppose you e-mail a joke to three friends on Monday. Each of those
friends sends the joke on to three of their friends on Tuesday. Each
person who receives the joke on Tuesday sends it to three more people
on Wednesday, and so on.
E-Mail Jokes
OOX
Monday
Tuesday
IM
15
2:10 PM
< >
Geometric Series 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, and 2187 form a geometric sequence
in which a = 1 and r = 3. The indicated sum of the numbers in the
sequence, 1+3+9+ 27 + 81 + 243 + 729 + 2187, is called a
geometric series.
To develop a formula for the sum of a geometric series, consider the series
given in the e-mail situation above. Multiply each term in the series by
the common ratio and subtract the result from the original series.
Sg=1+3+9+ 27 + 81 + 243 + 729 + 2187
(-) 35g = 3 + 9 + 27 +81 + 243 + 729 + 2187 + 6561
(1 – 3)Sg = 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 – 6561
first term in series
NARAR