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.
-2
Ithmetic Series
Main Ideas
• Find sums of
arithmetic series.
• Use sigma notation
New Vocabulary
series
arithmetic series
sigma notation
index of summation
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
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.
Study Tip
Indicated Sum
The sum of a series is
the resuk when the
terms of the series are
added. An indicated
sum is the expression
that illustrates the
series, which includes
the terms + or –
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.
5. represents the sum of the first n terms of a series. For example, 5, is the
sum of the first four terms.
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.
S. = 4 + 11 + 18 + 25 + 32 +39 +46 + 53 +60
(+) Sn = 60 +53 + 46 +39 + 32 + 25 + 18 + 11 + 4
25. = 64 +64 +64 +64 +64 +64 +64 +64 +64
2s, = 9(64)
Note that the sum had 9 terms
5. = 164)
The sum of the first and last terms of the series is 64.
An arithmetic series S, has n terms, and the sum of the first and last terms
is az + . Thus, the formula Sn = ( + 0,) represents the sum of any
arithmetic series.
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