8.2 notes.notebook
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Section 8.2 Arithmetic Sequences & Series
Learning Objectives:
1) Find the common difference (d) for an arithmetic sequence
2) Write the terms of an arithmetic sequence
3) Derive/apply the formula for the general term an of an arithmetic sequence
4) Derive/apply the recursive formula of an arithmetic sequence
5) Derive/apply the formula for the sum of the first n terms of an arithmetic sequence
Aug 12­4:32 PM
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8.2 notes.notebook
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Arithmetic sequence a sequence in which each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the common difference of the sequence (d).
d = an+1 ­ an, n ≥ 1
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1) Find the common difference (d) of the arithmetic sequence.
a. 5, 9, 13, 17, ...
b. 6, 1, ­4, ­9, ...
2) Write the first 4 terms of each arithmetic sequence. a. a1 = 13 d = ­4
b. a1 = ­8 d = 3
c. an = 15 ­ 2n
Aug 24­10:09 AM
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Recursive Definition of an Arithmetic Sequence
The recursive definition of an arithmetic sequence with the first term a1 and common difference d is:
3) Write the recursive formula for the given arithmetic sequence a. 5, 9, 13, 17, ...
b. 6, 1, ­4, ­9, ...
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General or nth Term of an Arithmetic Sequence
The general or nth term of an arithmetic sequence with the first term a1 and common difference d is:
3) Write a general formula for the given arithmetic sequence and
then find the 20th term..
a. 5, 9, 13, 17, ...
b. 6, 1, ­4, ­9, ...
Aug 12­4:51 PM
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12.2 Part 2 ­ Arithmetic Sequences
Obj. 4: Use the formula for the sum of the first n terms of an arithmetic sequence.
How might we find a formula for the sum of the first n terms?
Here, is the sum:
Now, written backwards:
a1
Sn =
+ (a1 + d) + (a1 + 2d)
+ ... + an
+ (an ­ 2d)
+ ... + a1
Sn = an + (an ­ d)
Now, add the two equations together:
+
Sn =
a1
+ (a1 + d)
+ (a1 + 2d)
+ ... + an
Sn =
an
+ (an ­ d)
+ (an ­ 2d)
+ ... + a1
2Sn = (a1 + an) + (a1 + an) +
(a1 + an)
+ ... + (a1 + an)
n times
2Sn = n(a1 + an)
2Sn = n(a1 + an)
2
2
The sum, Sn, of the first n terms of an arithmetic sequence is given by:
Jun 9­11:26 AM
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An arithmetic series is a sum of the first n terms of an arithmetic sequence.
The formula for an arithmetic series is given by:
where n = the number of terms
a1 = the first term an = the nth term Aug 12­5:14 PM
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The formula for an arithmetic series is given by:
or:
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4) Find the sum of the first 10 terms of 5, 9, 13, 17, ... Aug 13­8:37 AM
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Use the formula for the sum of the first n terms to find the following summation.
Jun 9­12:05 PM
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Aug 13­8:48 AM
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8.2 Homework:
pp 669 ­ 71: 9 ­ 29 eoo, 37 ­ 45 eoo,
47 ­ 51 ODDS, 59, 63, 65, 67
Aug 12­5:29 PM
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