1. Sequences, Series, and Probability9
Sequences, Series, and Probability
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2. Arithmetic Sequences and Partial Sums 9.2
Copyright © Cengage Learning. All rights reserved.

3. Objectives
Recognize, write, and find the nth terms of arithmetic sequences.
Find nth partial sums of arithmetic sequences.
Use arithmetic sequences to model and solve real-life problems.

4. Arithmetic Sequences

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

6. Example 1 – Examples of Arithmetic Sequences
a. The sequence whose n th term is 4n + 3 is arithmetic. For this sequence, the common difference between consecutive terms is 4. 7, 11, 15, 19, , 4n + 3, b. The sequence whose nth term is 7 – 5n is arithmetic. For this sequence, the common difference between consecutive terms is – 5. 2, –3, – 8, –13, , 7 – 5n, . . .
Begin with n = 1.
11 – 7 = 4
Begin with n = 1.
–3 – 2 = –5

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

8. Arithmetic Sequences
The sequence 1, 4, 9, 16, , whose n th term is n2, is not arithmetic. The difference between the first two terms is a2 – a1 = 4 – 1 = 3 but the difference between the second and third terms is a3 – a2 = 9 – 4 = 5. The nth term of an arithmetic sequence can be derived from the following pattern. a1 = a1 a2 = a1 + d
1st term.
2nd term.

9. Arithmetic Sequences
a3 = a1 + 2d a4 = a1 + 3d a5 = a1 + 4d an = a1 + (n – 1)d
3rd term.
4th term.
5th term.
nth term.

10. Arithmetic Sequences
The following definition summarizes this result.

11. Example 2 – Finding the nth Term
Find a formula for the n th term of the arithmetic sequence whose common difference is 3 and whose first term is 2. Solution: You know that the formula for the n th term is of the form an = a1 + ( n – 1)d. Moreover, because the common difference is d = 3 and the first term is a1 = 2, the formula must have the form an = 2 + 3(n – 1).
Substitute 2 for a 1 and 3 for d.

12. Example 2 – Solution
cont’d
So, the formula for the nth term is an = 3n – 1. The sequence therefore has the following form. 2, 5, 8, 11, 14, , 3n – 1, The figure below shows a graph of the first 15 terms of the sequence.

13. Example 2 – Solution
cont’d
Notice that the points lie on a line. This makes sense because an is a linear function of n. In other words, the terms “arithmetic” and “linear” are closely connected.

14. Arithmetic Sequences
When you know the n th term of an arithmetic sequence and you know the common difference of the sequence, you can find the (n + 1)th term by using the recursion formula an + 1 = an + d. With this formula, you can find any term of an arithmetic sequence, provided that you know the preceding term. For instance, when you know the first term, you can find the second term. Then, knowing the second term, you can find the third term, and so on.
Recursion formula

15. The Sum of a Finite Arithmetic Sequence

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

17. Example 5 – Sum of a Finite Arithmetic Sequence
Find the sum: 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 = (a1 + an)
Sum of a finite arithmetic sequence

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

19. The Sum of a Finite Arithmetic Sequence
The sum of the first n terms of an infinite sequence is the n th partial sum. The n th partial sum can be found by using the formula for the sum of a finite arithmetic sequence.

20. Application

21. Example 9 – Total Sales
A small business sells $10,000 worth of skin care products during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 9 years. Assuming that this goal is met, find the total sales during the first 10 years this business is in operation.

22. Example 9 – Solution
The annual sales form an arithmetic sequence in which a1 = 10,000 and d = So, an = 10, (n – 1) and the nth term of the sequence is an = 7500n Therefore, the 10th term of the sequence is a10 = 7500(10)

23. Example 9 – Solution
cont’d
= 77,500 The sum of the first 10 terms of the sequence is
See figure.
nth partial sum formula
Substitute 10 for n, 10,000 for a1, and 77,500 for a10.

24. Example 9 – Solution
cont’d
= 5(87,500) = 437,500. So, the total sales for the first 10 years will be $437,500.
Simplify.
Multiply.