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All courses Algebra I · F-BF.2 17 of 76
Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.

Write a recursive arithmetic sequence rule from terms

Problem
Write a recursive arithmetic-sequence rule for the terms \(5,~8,~11,~14\).
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Hint

Use the first listed term as the starting value, then compare consecutive terms to find the repeated change.

A recursive arithmetic rule names the first term and then adds the common difference each step.

Solution walkthrough

01

Set the initial term

\[\text{first}~\text{listed}~\text{term}~=~5~->~a_1~=~5\]

A recursive sequence needs an initial condition so its later terms are uniquely determined.

02

Compute consecutive differences

\[8~-~5~=~3;~11~-~8~=~3;~14~-~11~=~3\]

Every difference is the same, so the sequence is arithmetic with common difference +3.

03

Write the recursive step

\[a_n~=~a_{n-1}~+~3~\text{for}~n~\ge~2\]

After the first term, each new term is obtained by adding 3 to the immediately previous term.

04

State and verify the rule

\[a_1~=~5;~a_n~=~a_{n-1}~+~3\]

Starting at 5, the recurrence generates 8, then 11, then 14, exactly matching the source list.

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Another way

  1. Read the list as start at 5, then repeat add 3; translate those two instructions directly into initial and recursive rules.

!

Common mistake

Do not omit a₁ = 5. A recurrence without a starting value describes many possible sequences rather than this one.

Solution walkthrough video