Growing Patterns Formula

Q: What do the symbols mean in the Growing Patterns formula?

$a_n$ is the $n$th term; $d$ is the common difference added at each step

The Formula

a_n = a_1 + (n - 1) \cdot d for a pattern growing by a constant d

When to use: Imagine stacking blocks in a staircase—each step is one block taller than the last. The pattern grows by a rule: +1 block per step. If the rule is +3, the staircase grows faster.

Quick Example

2, 5, 8, 11, 14, \ldots \quad (\text{rule: } +3) 1, 4, 9, 16, 25, \ldots \quad (\text{rule: } n^2)

Notation

a_n is the nth term; d is the common difference added at each step

What This Formula Means

A pattern where each term changes by a consistent rule, such as adding the same number each time.

Imagine stacking blocks in a staircase—each step is one block taller than the last. The pattern grows by a rule: +1 block per step. If the rule is +3, the staircase grows faster.

Worked Examples

Example 1

easy

A pattern starts: 3, 7, 11, 15, ... Find the next two terms and the rule.

Solution

1
Find the common difference: $7-3=4$, $11-7=4$, $15-11=4$.
2
The rule is: add 4 each time.
3
Next term: $15 + 4 = 19$.
4
Term after: $19 + 4 = 23$.

Answer

19, 23 (rule: add 4)

This is an arithmetic sequence with first term $a_1 = 3$ and common difference $d = 4$. Formula: $a_n = 3 + (n-1) \times 4$.

Example 2

medium

The first term of a pattern is 5 and it grows by 6 each time. Using $a_n = a_1 + (n-1)d$, find the 10th term.

Example 3

medium

Find the 10th term of the pattern: 3, 7, 11, 15, ...

Common Mistakes

Assuming all growing patterns add the same amount (some multiply or follow other rules)
Looking only at consecutive terms instead of the relationship to the position number
Confusing the difference between terms with the terms themselves

Why This Formula Matters

Growing patterns lead directly to algebra and functions, where rules describe how quantities change.

Frequently Asked Questions

What is the Growing Patterns formula?

A pattern where each term changes by a consistent rule, such as adding the same number each time.

How do you use the Growing Patterns formula?

Imagine stacking blocks in a staircase—each step is one block taller than the last. The pattern grows by a rule: +1 block per step. If the rule is +3, the staircase grows faster.