Learn on PengiCalifornia Reveal Math, Algebra 1Unit 8: Exponential Functions

8-6 Recursive Formulas

In this Grade 9 lesson from California Reveal Math Algebra 1, students learn to write and evaluate recursive formulas for arithmetic and geometric sequences, using the notation a_n = a_(n-1) + d for arithmetic sequences and a_n = r · a_(n-1) for geometric sequences. Students practice identifying common differences and common ratios to determine sequence type, then construct recursive formulas from lists of terms and graphs. The lesson also contrasts recursive formulas with explicit formulas, helping students understand two distinct methods for finding the nth term of a sequence.

Section 1

Introduction to Recursive Rules for Sequences

Property

A recursive formula defines each term of a sequence using one or more previous terms. For single-step sequences, it has two parts:

The initial condition: the value of the first term, $a_1$
The recursive rule: a formula expressing $a_n$ in terms of the immediately preceding term $a_{n-1}$

Section 2

Writing Recursive Rules for Arithmetic Sequences

Property

For an arithmetic sequence, the recursive rule is $a_n = a_{n-1} + d$ where $d$ is the common difference, along with the initial term $a_1$ .

Examples

Section 3

Recursive Formula for Geometric Sequences (Property Form)

Property

A recursive formula for a geometric sequence expresses each term in relation to the previous term:

a_n = r \cdot a_{n-1}

where

r

is the common ratio and

a_{n-1}

is the previous term.

Examples

Section 4

Evaluating Terms Iteratively from a Recursive Rule

Property

To generate terms from a recursive formula, start with the given initial term $a_1$ and repeatedly apply the recurrence relation to find each successive term:

a_1 \text{ (given)}, \quad a_2 = f(a_1), \quad a_3 = f(a_2), \quad a_4 = f(a_3), \quad \ldots

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Introduction to Recursive Rules for Sequences

Property

A recursive formula defines each term of a sequence using one or more previous terms. For single-step sequences, it has two parts:

The initial condition: the value of the first term, $a_1$
The recursive rule: a formula expressing $a_n$ in terms of the immediately preceding term $a_{n-1}$

Section 2

Writing Recursive Rules for Arithmetic Sequences

Property

For an arithmetic sequence, the recursive rule is $a_n = a_{n-1} + d$ where $d$ is the common difference, along with the initial term $a_1$ .

Examples

Section 3

Recursive Formula for Geometric Sequences (Property Form)

Property

A recursive formula for a geometric sequence expresses each term in relation to the previous term:

a_n = r \cdot a_{n-1}

where

r

is the common ratio and

a_{n-1}

is the previous term.

Examples

Section 4

Evaluating Terms Iteratively from a Recursive Rule

Property

To generate terms from a recursive formula, start with the given initial term $a_1$ and repeatedly apply the recurrence relation to find each successive term:

a_1 \text{ (given)}, \quad a_2 = f(a_1), \quad a_3 = f(a_2), \quad a_4 = f(a_3), \quad \ldots

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Introduction to Recursive Rules for Sequences

Property

A recursive formula defines each term of a sequence using one or more previous terms. For single-step sequences, it has two parts:

The initial condition: the value of the first term, $a_1$
The recursive rule: a formula expressing $a_n$ in terms of the immediately preceding term $a_{n-1}$

Section 2

Writing Recursive Rules for Arithmetic Sequences

Property

For an arithmetic sequence, the recursive rule is $a_n = a_{n-1} + d$ where $d$ is the common difference, along with the initial term $a_1$ .

Examples

Section 3

Recursive Formula for Geometric Sequences (Property Form)

Property

A recursive formula for a geometric sequence expresses each term in relation to the previous term:

a_n = r \cdot a_{n-1}

where

r

is the common ratio and

a_{n-1}

is the previous term.

Examples

Section 4

Evaluating Terms Iteratively from a Recursive Rule

Property

To generate terms from a recursive formula, start with the given initial term $a_1$ and repeatedly apply the recurrence relation to find each successive term:

a_1 \text{ (given)}, \quad a_2 = f(a_1), \quad a_3 = f(a_2), \quad a_4 = f(a_3), \quad \ldots