Learn on PengiBig Ideas Math, Algebra 2Chapter 8: Sequences and Series

Lesson 5: Using Recursive Rules with Sequences

In this Grade 8 lesson from Big Ideas Math Algebra 2, Chapter 8, students learn how to evaluate and write recursive rules for sequences, including arithmetic sequences using the recursive equation a_n = a_{n-1} + d and geometric sequences using a_n = r · a_{n-1}. Students also practice translating between recursive and explicit rules and apply recursive rules to real-life problems. Special sequences such as the Fibonacci sequence and factorial sequences are introduced as examples of recursive patterns that are neither strictly arithmetic nor geometric.

Section 1

Recursive Rules and Equations

Property

A recursive rule defines a sequence by giving one or more initial terms and a recursive equation that expresses each subsequent term in relation to previous terms. The general form is: an=f(an−1,an−2,...)a_n = f(a_{n-1}, a_{n-2}, ...) where ff is some function of previous terms.

Examples

Section 2

Writing Recursive Rules for Arithmetic Sequences

Property

For an arithmetic sequence, the recursive rule is an=an−1+da_n = a_{n-1} + d where dd is the common difference, along with the initial term a1a_1.

Examples

Section 3

Writing Recursive Rules for Geometric Sequences

Property

A recursive rule for a geometric sequence has the form an=r×an−1a_n = r \times a_{n-1} where rr is the common ratio and a1a_1 is the first term.

Examples

Book overview

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Chapter 8: Sequences and Series

  1. Lesson 1

    Lesson 1: Defining and Using Sequences and Series

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  2. Lesson 2

    Lesson 2: Analyzing Arithmetic Sequences and Series

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  3. Lesson 3

    Lesson 3: Analyzing Geometric Sequences and Series

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Lesson overview

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Section 1

Recursive Rules and Equations

Property

A recursive rule defines a sequence by giving one or more initial terms and a recursive equation that expresses each subsequent term in relation to previous terms. The general form is: an=f(an−1,an−2,...)a_n = f(a_{n-1}, a_{n-2}, ...) where ff is some function of previous terms.

Examples

Section 2

Writing Recursive Rules for Arithmetic Sequences

Property

For an arithmetic sequence, the recursive rule is an=an−1+da_n = a_{n-1} + d where dd is the common difference, along with the initial term a1a_1.

Examples

Section 3

Writing Recursive Rules for Geometric Sequences

Property

A recursive rule for a geometric sequence has the form an=r×an−1a_n = r \times a_{n-1} where rr is the common ratio and a1a_1 is the first term.

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 8: Sequences and Series

  1. Lesson 1

    Lesson 1: Defining and Using Sequences and Series

    LearnPractice
  2. Lesson 2

    Lesson 2: Analyzing Arithmetic Sequences and Series

    LearnPractice
  3. Lesson 3

    Lesson 3: Analyzing Geometric Sequences and Series

    LearnPractice