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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 5.7 Arithmetic and Geometric Sequences

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Section 5.7 Arithmetic and Geometric Sequences. What You Will Learn. Arithmetic Sequences Geometric Sequences. Sequences. A sequence is a list of numbers that are related to each other by a rule. The terms are the numbers that form the sequence. Arithmetic Sequence. - PowerPoint PPT Presentation

Slide 1Section 5.7

What You Will Learn

Sequences

A sequence is a list of numbers that are related to each other by a rule.

The terms are the numbers that form the sequence.

5.7-*

Arithmetic Sequence

An arithmetic sequence is a sequence in which each term after the first term differs from the preceding term by a constant amount.

The common difference, d, is the amount by which each pair of successive terms differs.

To find the difference, simply subtract any term from the term that directly follows it.

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Example 2: An Arithmetic Sequence with a Negative Difference

Write the first five terms of the arithmetic sequence with first term 9 and a common difference of –4.

Solution

9, 5, 1, –3, –7

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General or nth Term of an Arithmetic Sequence

For an arithmetic sequence with first term a1 and common difference d, the general or nth term can be found using the following formula.

an = a1 + (n – 1)d

Example 3: Determining the 12th Term of an Arithmetic Sequence

Determine the twelfth term of the arithmetic sequence whose first term is –5 and whose common difference is 3.

Solution

an = a1 + (n – 1)d

a12 = –5 + (12 – 1)3

Example 4: Determining an Expression for the nth Term

Write an expression for the general or nth term, an, for the sequence

1, 6, 11, 16,…

an = a1 + (n – 1)d

Sum of the First n Terms of an Arithmetic Sequence

The sum of the first n terms of an arithmetic sequence can be found with the following formula where a1 represents the first term and an represents the nth term.

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Example 5: Determining the Sum of an Arithmetic Sequence

Determine the sum of the first 25 even natural numbers.

Solution

Substitute a1 = 2, a25 = 50, n = 25 into the formula

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Example 5: Determining the Sum of an Arithmetic Sequence

Solution

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Geometric Sequences

A geometric sequence is one in which the ratio of any term to the term that directly precedes it is a constant.

This constant is called the common ratio, r.

r can be found by taking any term except the first and dividing it by the preceding term.

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Example 6: The First Five Terms of a Geometric Sequence

Write the first five terms of the geometric sequence whose first term, a1, is 5 and whose common ratio, r, is 2.

Solution

5, 10, 20, 40, 80

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General or nth Term of a Geometric Sequence

For a geometric sequence with first term a1 and common ratio r, the general or nth term can be found using the following formula.

an = a1r n–1

Example 7: Determining the 12th Term of a Geometric Sequence

Determine the twelfth term of the geometric sequence whose first term is –4 and whose common ratio is 2.

Solution

an = a1r n–1

a12 = –4 • 212–1

= –4 • 211 = –4 • 2048

Example 8: Determining an Expression for the nth Term

Write an expression for the general or nth term, an, for the sequence

2, 6, 18, 54,…

an = a1r n–1

Sum of the First n Terms of an Geometric Sequence

The sum of the first n terms of an geometric sequence can be found with the following formula where a1 represents the first term and r represents the common ratio.

5.7-*

Example 9: Determining the Sum of an Geometric Sequence

Determine the sum of the first five terms in the geometric sequence whose first term is 4 and whose common ratio is 2.

Solution

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Example 5: Determining the Sum of an Arithmetic Sequence

Solution

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What You Will Learn

Sequences

A sequence is a list of numbers that are related to each other by a rule.

The terms are the numbers that form the sequence.

5.7-*

Arithmetic Sequence

An arithmetic sequence is a sequence in which each term after the first term differs from the preceding term by a constant amount.

The common difference, d, is the amount by which each pair of successive terms differs.

To find the difference, simply subtract any term from the term that directly follows it.

5.7-*

Example 2: An Arithmetic Sequence with a Negative Difference

Write the first five terms of the arithmetic sequence with first term 9 and a common difference of –4.

Solution

9, 5, 1, –3, –7

5.7-*

General or nth Term of an Arithmetic Sequence

For an arithmetic sequence with first term a1 and common difference d, the general or nth term can be found using the following formula.

an = a1 + (n – 1)d

Example 3: Determining the 12th Term of an Arithmetic Sequence

Determine the twelfth term of the arithmetic sequence whose first term is –5 and whose common difference is 3.

Solution

an = a1 + (n – 1)d

a12 = –5 + (12 – 1)3

Example 4: Determining an Expression for the nth Term

Write an expression for the general or nth term, an, for the sequence

1, 6, 11, 16,…

an = a1 + (n – 1)d

Sum of the First n Terms of an Arithmetic Sequence

The sum of the first n terms of an arithmetic sequence can be found with the following formula where a1 represents the first term and an represents the nth term.

5.7-*

Example 5: Determining the Sum of an Arithmetic Sequence

Determine the sum of the first 25 even natural numbers.

Solution

Substitute a1 = 2, a25 = 50, n = 25 into the formula

5.7-*

Example 5: Determining the Sum of an Arithmetic Sequence

Solution

5.7-*

Geometric Sequences

A geometric sequence is one in which the ratio of any term to the term that directly precedes it is a constant.

This constant is called the common ratio, r.

r can be found by taking any term except the first and dividing it by the preceding term.

5.7-*

Example 6: The First Five Terms of a Geometric Sequence

Write the first five terms of the geometric sequence whose first term, a1, is 5 and whose common ratio, r, is 2.

Solution

5, 10, 20, 40, 80

5.7-*

General or nth Term of a Geometric Sequence

For a geometric sequence with first term a1 and common ratio r, the general or nth term can be found using the following formula.

an = a1r n–1

Example 7: Determining the 12th Term of a Geometric Sequence

Determine the twelfth term of the geometric sequence whose first term is –4 and whose common ratio is 2.

Solution

an = a1r n–1

a12 = –4 • 212–1

= –4 • 211 = –4 • 2048

Example 8: Determining an Expression for the nth Term

Write an expression for the general or nth term, an, for the sequence

2, 6, 18, 54,…

an = a1r n–1

Sum of the First n Terms of an Geometric Sequence

The sum of the first n terms of an geometric sequence can be found with the following formula where a1 represents the first term and r represents the common ratio.

5.7-*

Example 9: Determining the Sum of an Geometric Sequence

Determine the sum of the first five terms in the geometric sequence whose first term is 4 and whose common ratio is 2.

Solution

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Example 5: Determining the Sum of an Arithmetic Sequence

Solution

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