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All courses Math I · F-BF.2 17 of 59
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

Identify the first term

\[a_1~=~5\]

The first term in the list is the starting value for the recursive rule.

02

Find the common difference

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

Each term is 3 more than the term before it, so the common difference is 3.

03

Write the recursive step

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

After the first term, each new term comes from adding 3 to the previous term.

04

State the rule

\[a_1~=~5;~a_n~=~a_(n-1)~+~3\]

Together, the starting value and recursive step generate exactly the listed sequence.

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Another way

  1. Check by starting at 5 and adding 3 repeatedly: 5, 8, 11, 14.

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Common mistake

A common mistake is to write only aₙ = aₙ₋₁ + 3 and omit a₁ = 5. A recursive rule needs the starting term so the sequence is determined.