11­4 Arithmetic Series
April 27, 2009
11­4 Arithmetic Series
Objectives:
• Write and evaluate arithmetic series.
• Use summation notation.
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Find each sum.
1. 2 + 3.5 + 5 + 6.5 + 8
2. ­17 + (­13) + (­9) + (­5) + (­1) + 3
Write an explicit formula for each sequence.
3. 4, 6, 7, 10, 12,...
4. 1, 4, 7, 10, 13, 16,...
5. ­17, ­23, ­29, ­35,...
6. 10, 1, ­8, ­17,...
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Apr 21­4:17 PM
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Series: The expression for the sum of the terms of a sequence.
Finite Sequence
6, 9, 12, 15, 18
Finite Series
6 + 9 + 12 + 15 + 18
Infinite Sequence
3, 7, 11, 15,...
Infinite Series
3 + 7 + 11 +15 +...
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Example #1: Writing and Evaluating a Series
Use the finite sequence 2, 11, 20, 29, 38, 47.
a. Write the related series.
2 + 11 + 20 + 29 + 38 + 47
b. Evaluate the series.
2 + 11 + 20 + 29 + 38 + 47 = 147
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Arithmetic Series: A series whose terms form an arithmetic sequence.
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You can use the summation symbol Ʃ to write a series.
Then you can use limits to indicate how many terms you are adding.
Ʃ is the Greek capital
letter sigma, the
equivalent of the English
letter S (for summation).
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Example #3: Writing a Series in Summation Notation
Use summation notation to write the series 3 + 6 + 9 +... for 33 terms.
Step 1: Write the explicit formula.
an = a1 + d(n ­ 1)
an = 3 + 3(n ­ 1)
an = 3n
Step 2: Write in summation notation including the limits of n.
33
3n
n=1
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Example #4: Writing a Series in Summation Notation
Use summation notation to write the series 3 + 8 + 13 + 18... for n = 9.
Step 1: Write the explicit formula.
an = a1 + d(n ­ 1)
an = 3 + 5(n ­ 1)
an = 5n ­ 2
Step 2: Write in summation notation including the limits of n.
9
(5n ­ 2)
n=1
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To expand a series from summation notation,
you can substitute each value of n into the
explicit formula and add the results.
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Example #5: Finding the Sum of a Series
3
Use the series
(5n + 1).
n=1
a. Find the number of terms in the series.
3 terms
n = 1, 2, 3
b. Find the first and last terms of the series.
a1 = 5(1) + 1 = 6
an = 5(3) + 1 = 16
c. Evaluate the series.
3
(5n + 1) = (5(1) + 1) + (5(2) + 1) + (5(3) + 1)
n=1
= 6 + 11 + 16
= 33
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Example #6: For each sum, find the number of terms, the first term,
and the last term. Then evaluate each series.
n
Sn =
4
10
a.
(n ­ 3)
n=1
b.
1
2
( n + 1)
n=1
2
(a1 + an)
5
c.
n2
n=2
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Homework:
page 622 (2 ­ 32 even, 37, 40)
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