Number Patterns and Algebra

A pattern is a sequence of numbers or shapes that follows a consistent rule from one term to the next. In algebra, the goal is not just to notice the pattern, it's to describe that rule precisely, usually as an expression in terms of the position number \(n\) (sometimes written as the 1st term, 2nd term, and so on). Once you have the rule, you can find any term in the sequence, even one far beyond what's shown, without writing out every step in between.

Patterns show up constantly in algebra: they're the bridge between counting and writing formal expressions. Once you can describe a pattern with an expression, you're ready to evaluate algebraic expressions and eventually solve equations built from them.

Most patterns in this topic fall into a few common categories:

• Arithmetic (number) patterns: each term is found by adding or subtracting the same fixed amount, called the common difference. Example: 2, 5, 8, 11, ...
• Geometric patterns: each term is found by multiplying (or dividing) by the same fixed amount, called the common ratio. Example: 3, 6, 12, 24, ...
• Shape (visual) patterns: a figure made of dots, tiles, or squares grows in a consistent way from one stage to the next.
• Special sequences: some patterns follow their own famous rule, such as the Fibonacci sequence, where each term is the sum of the two terms before it: 1, 1, 2, 3, 5, 8, 13, ...

To find a rule, first check whether the difference between consecutive terms is constant (arithmetic) or whether the ratio is constant (geometric). Once you know which type it is, connect the term number \(n\) to the term's value.

Worked example. Consider the sequence 2, 5, 8, 11, 14, ...

• Term 1 = 2, Term 2 = 5, Term 3 = 8. The difference between consecutive terms is always \(3\), so this is arithmetic.
• Compare each term to \(3n\): when \(n = 1\), \(3n = 3\), but the term is \(2\), which is \(1\) less. When \(n = 2\), \(3n = 6\), and the term is \(5\), again \(1\) less.
• So the rule is \(a_n = 3n - 1\), where \(a_n\) is the value of the \(n\)th term.

You can check the rule against every given term before trusting it: for \(n = 4\), \(3(4) - 1 = 11\), which matches. That confirms the rule works.

Because the terms of a sequence are really outputs of an expression evaluated at whole numbers, writing and checking a pattern rule uses the same skills as evaluating algebraic expressions: substitute a value for \(n\) and simplify.

The real power of finding a rule is that you can jump ahead. For the sequence above, what is the 20th term? Instead of listing 20 numbers, just substitute \(n = 20\) into the rule:

\(a_{20} = 3(20) - 1 = 60 - 1 = 59\)

This is also how "complete the pattern" and "find the missing term" problems work: once the rule is known, plug in the position of the missing term to fill in the gap.

Shape patterns describe growth visually, but the same idea applies: count how many tiles, dots, or squares are added at each stage, and turn that into a rule based on the stage number.

Here the count of squares at each stage is 1, 3, 6, ... The differences between terms are 2, then 3, then (continuing the pattern) 4, so this isn't a simple arithmetic pattern with a constant difference, it's a pattern where the amount added itself grows by 1 each time. Recognizing when a pattern isn't purely arithmetic or geometric is just as important as recognizing when it is.

One of the most famous special sequences is the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, ... Each term is the sum of the two terms immediately before it. Unlike arithmetic or geometric patterns, there's no single constant difference or ratio, the rule instead relates each term to the two terms that came before it. This sequence also generates the well-known Fibonacci spiral, built from squares whose side lengths follow these same numbers.

Writing a pattern rule as an algebraic expression is often the first step toward writing and solving an equation. For instance, if you know a pattern's rule is \(a_n = 3n - 1\) and you want to know which term equals 41, you set up the equation \(3n - 1 = 41\) and solve for \(n\). This connects directly to skills like solving one-step equations and, for rules with more steps, two-step equations. Practicing pattern rules now builds the exact habits, substituting values, isolating a variable, checking a solution, that carry through the rest of algebra.

Related lessons

• Algebra Help