Learn on PengiBig Ideas Math, Algebra 1Chapter 6: Exponential Functions and Sequences

Lesson 7: Recursively Defi ned Sequences

Property A sequence is a function whose domain is the counting numbers. A sequence can also be seen as an ordered list of numbers and each number in the list is a term. A sequence may have an infinite number of terms (infinite sequence) or a finite number of terms (finite sequence). The notation $a n$ represents the $n$th term of the sequence.

Section 1

Sequences

Property

A sequence is a function whose domain is the counting numbers.
A sequence can also be seen as an ordered list of numbers and each number in the list is a term.
A sequence may have an infinite number of terms (infinite sequence) or a finite number of terms (finite sequence).
The notation $a_n$ represents the $n$ th term of the sequence.

Examples

Write the first four terms of the sequence with general term $a_n = 3n + 2$ . The terms are $a_1 = 3(1)+2=5$ , $a_2 = 3(2)+2=8$ , $a_3 = 3(3)+2=11$ , and $a_4 = 3(4)+2=14$ . The sequence is $5, 8, 11, 14, \ldots$ .

Write the first four terms of the sequence with general term $a_n = (-1)^n(n+1)$ . The terms are $a_1 = (-1)^1(1+1)=-2$ , $a_2 = (-1)^2(2+1)=3$ , $a_3 = (-1)^3(3+1)=-4$ , and $a_4 = (-1)^4(4+1)=5$ . The sequence is $-2, 3, -4, 5, \ldots$ .

Section 2

General term of a sequence

Property

The general term of the sequence is found from the formula for writing the $n$ th term of the sequence.
The $n$ th term of the sequence, $a_n$ , is the term in the $n$ th position where $n$ is a value in the domain.

Examples

Find a general term for the sequence $5, 10, 15, 20, 25, \ldots$ . Each term is 5 times its position number, $n$ . So, the general term is $a_n = 5n$ .

Find a general term for the sequence $1, -2, 3, -4, 5, \ldots$ . The numbers are the position, $n$ , but the signs alternate, starting with positive. So, the general term is $a_n = (-1)^{n+1}n$ .

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Sequences

Property

Examples

Write the first four terms of the sequence with general term $a_n = 3n + 2$ . The terms are $a_1 = 3(1)+2=5$ , $a_2 = 3(2)+2=8$ , $a_3 = 3(3)+2=11$ , and $a_4 = 3(4)+2=14$ . The sequence is $5, 8, 11, 14, \ldots$ .

Write the first four terms of the sequence with general term $a_n = (-1)^n(n+1)$ . The terms are $a_1 = (-1)^1(1+1)=-2$ , $a_2 = (-1)^2(2+1)=3$ , $a_3 = (-1)^3(3+1)=-4$ , and $a_4 = (-1)^4(4+1)=5$ . The sequence is $-2, 3, -4, 5, \ldots$ .

Section 2

General term of a sequence

Property

Examples

Find a general term for the sequence $5, 10, 15, 20, 25, \ldots$ . Each term is 5 times its position number, $n$ . So, the general term is $a_n = 5n$ .

Find a general term for the sequence $1, -2, 3, -4, 5, \ldots$ . The numbers are the position, $n$ , but the signs alternate, starting with positive. So, the general term is $a_n = (-1)^{n+1}n$ .

Overview

Lesson overview

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

Overview

Section 1

Sequences

Property

Examples

Write the first four terms of the sequence with general term $a_n = 3n + 2$ . The terms are $a_1 = 3(1)+2=5$ , $a_2 = 3(2)+2=8$ , $a_3 = 3(3)+2=11$ , and $a_4 = 3(4)+2=14$ . The sequence is $5, 8, 11, 14, \ldots$ .

Write the first four terms of the sequence with general term $a_n = (-1)^n(n+1)$ . The terms are $a_1 = (-1)^1(1+1)=-2$ , $a_2 = (-1)^2(2+1)=3$ , $a_3 = (-1)^3(3+1)=-4$ , and $a_4 = (-1)^4(4+1)=5$ . The sequence is $-2, 3, -4, 5, \ldots$ .

Section 2

General term of a sequence

Property

Examples

Find a general term for the sequence $5, 10, 15, 20, 25, \ldots$ . Each term is 5 times its position number, $n$ . So, the general term is $a_n = 5n$ .

Find a general term for the sequence $1, -2, 3, -4, 5, \ldots$ . The numbers are the position, $n$ , but the signs alternate, starting with positive. So, the general term is $a_n = (-1)^{n+1}n$ .