Download Chapter 11 Section 2 - Canton Local Schools

Chapter 11
Further Topics
in Algebra
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All rights reserved
© 2010 Pearson Education, Inc. All rights reserved
1
SECTION 11.2 Arithmetic Sequences; Partial Sums
OBJECTIVES
1
2
Identify an arithmetic sequence and find its
common difference.
Find the sum of the first n terms of an
arithmetic sequence.
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2
DEFINITION OF AN ARITHMETIC SEQUENCE
The sequence
a1, a2, a3, a4, … , an, …
is an arithmetic sequence, or an arithmetic
progression if there is a number d such that
each term in the sequence except the first is
obtained from the preceding term by adding
d to it. The number d is called the common
difference of the arithmetic sequence. We
have
d = an + 1 – an, n ≥ 1.
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RECURSIVE DEFINITION OF AN
ARITHMETIC SEQUENCE
An arithmetic sequence
a1, a2, a3, a4, … , an, …
can be defined recursively. The recursive
formula
an + 1 = an + d for n ≥ 1
defines an arithmetic sequence with first
term a1 and common difference d.
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4
nTH TERM OF AN ARITHMETIC
SEQUENCE
If a sequence a1, a2, a3, … is an arithmetic
sequence, then its nth term, an, is given by
an = a1 + (n – 1)d,
where a1 is the first term and d is the
common difference.
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EXAMPLE 3
Finding the Common Difference of an
Arithmetic Sequence
Find the common difference d and the nth term
an of an arithmetic sequence whose 5th term is
15 and whose 20th term is 45.
Solution
an  a1   n  1 d
an  a1   n  1 d
45  a1   20  1 d
15  a1   5  1 d
45  a1  19d
15  a1  4d
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EXAMPLE 3
Finding the Common Difference of an
Arithmetic Sequence
Solution continued
45  a1  19d
Solving the system of equations
15  a1  4d
gives a1 = 7 and d = 2.
an  a1   n  1 d
an  7   n  1 2
an  7  2n  2  2n  5
The nth term is given by an = 2n + 5, n ≥ 1.
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SUM OF n TERMS OF AN
ARITHMETIC SEQUENCE
Let a1, a2, a3, … an be the first n terms of an
arithmetic sequence with common difference
d. The sum Sn of these n terms is given by
 a1  an 
Sn  n 
,
 2 
where an = a1 + (n – 1)d.
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EXAMPLE 4
Finding the Sum of Terms of a Finite
Arithmetic Sequence
Find the sum of the arithmetic sequence of
numbers:
1 + 4 + 7 + … + 25
Solution
Arithmetic sequence with a1 = 1 and d = 3.
First find the number of terms.
an  a1   n  1 d
25  1   n  1 3
24   n  1 3
8  n 1
n9
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EXAMPLE 4
Finding the Sum of Terms of a Finite
Arithmetic Sequence
Solution continued
 a1  an 
Sn  n 

 2 
 1  25 
S9  9 

 2 
 9 13  117
Thus 1 + 4 + 7 + … + 25 = 117.
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EXAMPLE 5
Calculating the Distance Traveled by a
Freely Falling Object
The terms of the arithmetic sequence 16, 48,
80, 112,… give the number of feet that freely
falling space junk falls in successive seconds.
For this sequence, find the following:
a. The common difference d
b. The nth term
c. The distance the object travels in ten seconds
Solution
a. The common difference
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EXAMPLE 5
Calculating the Distance Traveled by a
Freely Falling Object
Solution continued
b.
c. First find a10.
The object falls
1600 ft in 10 s.
Then use the formula.
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