Growing Patterns Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Growing Patterns.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A growing pattern is a sequence where each term increases by following a consistent rule, such as adding the same number each time (2, 5, 8, 11,...) or multiplying by a constant factor (3, 6, 12, 24,...). Recognizing the rule lets you predict any term in the sequence.

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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A growing pattern increases by a consistent rule each term — adding a fixed amount or multiplying by a fixed factor — so you can predict any term.

Common stuck point: The procedure for growing patterns is the easy part; the trap is assuming the rule is always addition. Asking "Is the change between consecutive terms a constant amount or a constant factor?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the change between consecutive terms a constant amount or a constant factor?

Worked Examples

Example 1

easy

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

Answer

19, 23 (rule: add 4)

First step

Find the common difference:

7-3=4

11-7=4

15-11=4

Full solution

2
The rule is: add 4 each time.
3
Next term: $15 + 4 = 19$ .
4
Term after: $19 + 4 = 23$ .

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, ...

Example 4

medium

A pattern starts at

4

and grows by

3

each step. Find the 20th term using

a_n = a_1 + (n-1)d

Example 5

medium

A geometric pattern starts

3, 6, 12, 24, \dots

. Find the 7th term.

Example 6

medium

Triangular numbers:

1, 3, 6, 10, 15, \dots

. Find the 8th term.

Example 7

hard

Find the 25th term of

7, 12, 17, 22, \dots

Example 8

hard

A staircase has

1

block in row

1

3

in row

2

5

in row

3

, and so on (odd numbers). How many blocks are in the first

10

rows total?

Example 9

hard

The sum of the first

n

positive integers is given by

S_n = \tfrac{n(n+1)}{2}

. Use it to compute

1 + 2 + \cdots + 100

Example 10

challenge

The Fibonacci sequence starts

1, 1, 2, 3, 5, 8, \dots

where each term is the sum of the previous two. Find the 10th term.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A sequence goes: 2, 9, 16, 23, ... What are the next two terms?

Example 2

medium

A pattern has

a_1 = 10

and

d = 5

. Find the 8th term using

a_n = a_1 + (n-1)d

Example 3

easy

What is the next term:

2, 5, 8, 11, \dots

Example 4

easy

What is the next term:

3, 6, 12, 24, \dots

Example 5

easy

Describe the rule for

10, 20, 30, 40, \dots

Example 6

easy

Find the missing term:

4, 7, \underline{\ \ }, 13, 16

Example 7

easy

What is the 5th term of

1, 4, 7, 10, \dots

Example 8

easy

2, 4, 8, 16

an adding or multiplying pattern?

Example 9

easy

What is the next term:

5, 10, 15, 20, \dots

Example 10

easy

Find the next term:

1, 4, 9, 16, \dots

(square numbers).

Example 11

medium

Write a formula for the

n

th term of

7, 10, 13, 16, \dots

Example 12

medium

The pattern

2, 6, 18, 54

multiplies by 3. What is the 6th term?

Example 13

medium

A figure pattern uses

4, 7, 10, 13

tiles. How many tiles in figure 10?

Example 14

medium

Which term of

3, 7, 11, 15, \dots

equals 43?

Example 15

medium

Find the next term:

1, 2, 4, 7, 11, \dots

Example 16

medium

A savings plan: \$5 in week 1, then \$3 more each week. Total saved by end of week 4?

Example 17

medium

Identify the rule and 4th term:

80, 40, 20, \dots

Example 18

medium

2, 5, 10, 17, 26

, what rule based on position

n

generates the terms?

Example 19

challenge

Find a closed formula for

2, 5, 10, 17, \dots

and the 10th term.

Example 20

challenge

A pattern: 1 dot, then each figure adds a ring with 6 more dots than the last ring (6, 12, 18, ...). Total dots in figure 4?

Example 21

challenge

For

5, 8, 11, \dots

, prove the difference of any two consecutive terms is constant, and state it.

Example 22

medium

Find the 7th term of

2, 5, 8, 11, \dots

using the rule.

Example 23

easy

What are the next two terms of

4, 9, 14, 19, \dots

Example 24

easy

What is the rule for

5, 10, 20, 40, \dots

Example 25

easy

What is the 4th term of

7, 14, 21, \dots

Example 26

easy

Find the next term:

1, 4, 7, 10, 13, \dots

Example 27

easy

Describe the rule for

50, 45, 40, 35, \dots

Example 28

medium

Find the 12th term of

5, 9, 13, 17, \dots

Example 29

medium

A pattern has

a_1 = 100

and decreases by

7

each step. Find

a_{15}

Example 30

medium

Find the 10th term of

2, 6, 18, 54, \dots

Example 31

medium

A pattern goes

8, 11, 14, 17, \dots

. Which term equals

50

Example 32

hard

A pattern's 5th term is

22

and 9th term is

42

. Find the common difference.

Example 33

hard

A population doubles every

5

years, starting at

1000

. Find the population after

25

years.

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Related Concepts

Simple Patterns Addition Linear Relationship

Background Knowledge

These ideas may be useful before you work through the harder examples.

simple patternsaddition