Finding the Rule for a Sequence — Definition, Formula & Examples

Finding the rule for a sequence means identifying the pattern that connects the terms so you can write a formula to calculate any term in the sequence.

Given an ordered list of terms $a_1, a_2, a_3, \ldots$ , finding the rule means determining an explicit formula $a_n = f(n)$ or a recursive relation that generates each term from its position number $n$ or from preceding terms.

How It Works

Start by looking at the differences between consecutive terms. If those differences are constant, the sequence is arithmetic and the rule involves addition. If the ratios between consecutive terms are constant, the sequence is geometric and the rule involves multiplication. Once you identify the type, plug in values to build the formula. Always test your rule by checking that it produces the original terms.

Worked Example

Problem: Find the rule for the sequence: 5, 11, 17, 23, 29, ...

Find the differences: Subtract each term from the next to see if there is a constant difference.

11 - 5 = 6,\quad 17 - 11 = 6,\quad 23 - 17 = 6

Identify the type: The common difference is 6, so this is an arithmetic sequence with first term

a_1 = 5

and common difference

d = 6

Write the rule: Use the arithmetic sequence formula

a_n = a_1 + (n-1)d

and simplify.

a_n = 5 + (n-1)(6) = 6n - 1

Verify: Check:

a_1 = 6(1) - 1 = 5

a_4 = 6(4) - 1 = 23

. Both match the original sequence.

Answer: The rule is

a_n = 6n - 1

Why It Matters

Finding sequence rules is a core skill in algebra and standardized tests. It also appears in real contexts like predicting savings growth, modeling population changes, and writing simple computer programs that generate patterns.

Common Mistakes

Mistake: Assuming every sequence with growing terms is arithmetic.

Correction: Always check both the differences and the ratios. If differences aren't constant, try dividing consecutive terms to see if the ratio is constant (geometric). Some sequences follow neither pattern.