8.2
Arithmetic Sequences
and Partial Sums
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What You Should Learn
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Recognize, write, and find the nth terms of
arithmetic sequences.
Find nth partial sums of arithmetic sequences.
Make sure you write down this formula on slide
#8 and at least one example.
Use arithmetic sequences to model and solve
real-life problems. Make sure you write down
this formula on slide #10 and at least one
example.
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Arithmetic Sequences
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Arithmetic Sequences
A sequence whose consecutive terms have a common
difference is called an arithmetic sequence.
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Example 1 – Examples of Arithmetic Sequences
a. The sequence whose nth term is
4n + 3
is arithmetic.
The common difference between consecutive terms is 4.
7, 11, 15, 19, . . . , 4n + 3, . . .
Begin with n = 1.
11 – 7 = 4
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Example 1 – Examples of Arithmetic Sequences
cont’d
b. The sequence whose nth term is
7 – 5n
is arithmetic.
The common difference between consecutive terms is
–5.
2, –3, –8, –13, . . . , 7 – 5n, . . .
Begin with n = 1.
–3 – 2 = –5
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Example 1 – Examples of Arithmetic Sequences
cont’d
c. The sequence whose nth term is
is arithmetic.
The common difference between consecutive terms is
Begin with n = 1.
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Arithmetic Sequences
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The Sum of a Finite
Arithmetic Sequence
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The Sum of a Finite Arithmetic Sequence
There is a simple formula for the sum of a finite arithmetic
sequence.
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Example 5 – Finding the Sum of a Finite Arithmetic Sequence
Find the sum:
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19.
Solution:
To begin, notice that the sequence is arithmetic (with a
common difference of 2). Moreover, the sequence has 10
terms. So, the sum of the sequence is
Sn = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19
Formula for sum of an arithmetic sequence
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Example 5 – Solution
cont’d
Substitute 10 for n, 1 for a1, and 19 for an.
= 5(20)
Simplify.
= 100.
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The Sum of a Finite Arithmetic Sequence
The sum of the first n terms of an infinite sequence is called
the nth partial sum.
The nth partial sum of an arithmetic sequence can be found
by using the formula for the sum of a finite arithmetic
sequence which is on slide #10.
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Applications
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Example 7 – Total Sales
A small business sells $20,000 worth of sports memorabilia
during its first year. The owner of the business has set a
goal of increasing annual sales by $15,000 each year for
19 years. Assuming that this goal is met, find the total sales
during the first 20 years this business is in operation.
Solution:
The annual sales form an arithmetic sequence in which
a1 = 20000 and d = 15,000. So,
an = 20,000 + 15,000(n – 1)
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Example 7 – Solution
cont’d
and the nth term of the sequence is
an = 15,000n + 5000.
This implies that the 20th term of the sequence is
a20 = 15,000(20) + 5000
= 300,000 + 5000
= 305,000.
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Example 7 – Solution
cont’d
The sum of the first 20 terms of the sequence is
nth partial sum formula
Substitute 20 for n,
20,000 for a1, and
305,000 for an.
= 10(325,000)
Simplify.
= 3,250,000.
Simplify.
So, the total sales for the first 20 years are $3,250,000.
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