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All courses Algebra I · F-IF.3 21 of 76
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.

Identify the domain of a sequence

Problem
In this sequence, \(a_{1}\) is the first term, \(a_{2}\) is the second term, and so on. State the domain implied by this indexing.
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Hint

Focus on the first index shown. This sequence begins with a₁ and then continues with a₂, a₃, and so on.

The domain of a sequence is the set of allowed term numbers, and those term numbers are discrete inputs rather than every real number.

Solution walkthrough

01

Read the first index

\[a_1~\text{is}~\text{first}~->~\text{starting}~\text{input}~n~=~1\]

The prompt explicitly labels a₁ as the first term, so the sequence begins at term number 1 rather than 0.

02

Continue the index pattern

\[a_1,~a_2,~a_3,~...~->~n~=~1,~2,~3,~...\]

The phrase 'and so on' continues the discrete term numbers one integer at a time with no stated final term.

03

Classify the inputs

\[n~\text{is}~\text{an}~\text{integer}~\text{and}~n~\ge~1\]

Integers beginning at 1 are the positive integers; 0, negative integers, and noninteger inputs are excluded by this indexing.

04

State the domain

\[\text{positive}~\text{integers}:~n~=~1,~2,~3,~...\]

This exact discrete domain contains every term number implied by the source indexing.

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Another way

  1. List the inputs attached to the named terms: a₁ uses 1, a₂ uses 2, and the same pattern continues.

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Common mistake

Do not include 0 merely because some sequences start at a₀. This prompt explicitly starts with a₁.

Solution walkthrough video