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8.2 Arithmetic Sequences and Partial Sums

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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.

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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

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

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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

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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

= 5(20)

= 100.

cont’d

Substitute 10 for n, 1 for a1, and 19 for an.

Simplify.

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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

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

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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, and305,000 for an.

nth partial sum formula

Simplify.