This document discusses arithmetic sequences and their properties. It defines an arithmetic sequence as a sequence whose consecutive terms have a common difference. It provides examples of arithmetic sequences and their common differences. It also presents the formula for finding the nth partial sum of a finite arithmetic sequence and provides examples of using this formula to calculate sums. Finally, it demonstrates how to apply arithmetic sequences and partial sums to model and solve real-life problems such as calculating total sales over multiple years.

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8.2 Arithmetic Sequences
and Partial Sums

2. 2
What You Should Learn
• 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.

3. 3
Arithmetic Sequences

4. 4
Arithmetic Sequences
A sequence whose consecutive terms have a common
difference is called an arithmetic sequence.

5. 5
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

6. 6
Example 1 – Examples of Arithmetic Sequences
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
cont’d

7. 7
Example 1 – Examples of Arithmetic Sequences
c. The sequence whose nth term is
is arithmetic.
The common difference between consecutive terms is
Begin with n = 1.
cont’d

8. 8
Arithmetic Sequences

9. 9
The Sum of a Finite
Arithmetic Sequence

10. 10
The Sum of a Finite Arithmetic Sequence
There is a simple formula for the sum of a finite arithmetic
sequence.

11. 11
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

12. 12
Example 5 – Solution
= 5(20)
= 100.
cont’d
Substitute 10 for n, 1 for a1, and 19 for an.
Simplify.

13. 13
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.

14. 14
Applications

15. 15
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)

16. 16
Example 7 – Solution
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.
cont’d

17. 17
Example 7 – Solution
The sum of the first 20 terms of the sequence is
= 10(325,000)
= 3,250,000.
So, the total sales for the first 20 years are $3,250,000.
cont’d
Simplify.
Substitute 20 for n,
20,000 for a1, and
305,000 for an.
nth partial sum formula
Simplify.