What Comes Next?

Contents

Sequence Patterns

One of the troubles with finding “the next number” in a sequence is that mathematics is so powerful we can find more than one Rule that works. The next number in a sequence is determined by identifying the rule or structure that generates the terms. There isn’t one universal method—there are several families of patterns, and the key is figuring out which one fits the data best.

Arithmetic patterns (constant change)
- Look at how much the sequence increases or decreases each step.
- If the differences are constant → arithmetic sequence
  Example: 4, 7, 10, 13 → add 3 each time.
Geometric patterns (constant ratio)
- Look at how each term multiplies.
- If the ratio is constant → geometric sequence
  Example: 3, 6, 12, 24 → multiply by 2.
Differences of differences (quadratic or higher)
- If the first differences aren’t constant, check the second differences.
- Constant second differences → quadratic rule
  Example: 2, 5, 10, 17 → differences: 3, 5, 7 → second differences: 2.
Alternating or interleaved patterns
- Sometimes two sequences are woven together.
- Example: 2, 10, 4, 12, 6, 14 → odds +2, evens +2.
Functional rules (n-based formulas)
- Some sequences follow a formula like:
- Example: 2, 5, 10, 17, 26…
Pattern in digits or structure
- Not numerical operations, but visual or structural patterns.
- 1, 11, 21, 1211, 111221 → “look-and-say”
- 1, 2, 4, 8, 16 → powers of 2
Context-based or special sequences
- Some sequences come from known mathematical families:
  - Fibonacci
  - Prime numbers
  - Factorials
  - Triangular numbers
  - Pascal’s triangle diagonals
Underdetermination (the hidden truth)
- A finite list of numbers can fit infinitely many rules.
- So the “best” next number is the one that follows the simplest, most natural pattern consistent with the context.

A Practical Workflow for Identifying the Rule

Check differences (constant? increasing? alternating?)
Check ratios (constant? repeating?)
Check for alternating subsequences
Check for known families (squares, cubes, primes, Fibonacci)
Try to express the nth term
Consider context (school problem? puzzle? real-world data?)

A Non-obvious Insight

Many sequences used in classrooms are designed to reinforce number sense, not trick students. So the intended rule is usually:

simple
consistent
based on operations students already know

The core idea is: look for the simplest rule that explains all the terms so far. That rule might come from arithmetic, geometry, differences, repetition, or something more creative.

Examples

Option 1: Simple & Direct

Finding the Next Number in an Interleaved Pattern

Sequence: 4, 6, 12, 14, 28, 30, ___

What You Should Notice

The numbers jump up and down.
Some increases are small (+2), some are large (+6, +8, +14).
It doesn’t look like one single pattern.

Key Idea

This sequence is made of two smaller sequences woven together.

Think of it like two trains taking turns on the same track.

Step 1: Separate odd and even positions

Odd positions (1st, 3rd, 5th): 4, 12, 28
Even positions (2nd, 4th, 6th): 6, 14, 30

Step 2: Look at the differences

Odd: +8, +16
Even: +8, +16

Both subsequences grow by doubling the previous increase.

Step 3: Continue the pattern

Next increase = +32
Odd side: 28 + 32 = 60

Final Answer

The next number is 60.

Option 2: Visual “Two Trains”

Two Patterns, One Sequence

Track A (odd positions):

4 → 12 → 28 → ?
Differences: +8 → +16 → +32

Track B (even positions):

6 → 14 → 30
Differences: +8 → +16 → +32

Both tracks follow the same rule: Each jump doubles the previous jump.

So Track A’s next jump is +32:
28 + 32 = 60

Next number: 60

A quick check:
If both tracks follow the same “double the jump” rule, the sequence stays balanced and predictable.

Option 3: Guided Practice

How to Find the Next Number in a Pattern

Sequence: 4, 6, 12, 14, 28, 30, ___

1. Look for a simple pattern

Not constant. Not multiplying. Not repeating.

2. Try splitting into two patterns

Circle every other number:

Circle 1: 4, 12, 28
Circle 2: 6, 14, 30

3. Study each circle

Circle 1 differences: +8, +16
Circle 2 differences: +8, +16

Both circles follow the same rule:
Add a number that doubles each time.

4. Continue the rule

Next jump = 32
28 + 32 = 60

Answer: 60

Option 4: Use a Next Number in the Sequence Calculator

Using the Find next number in the sequence calculator at AtoZmath.com

References

Brilliant. “What Comes Next?” Accessed February 26, 2026. https://brilliant.org/wiki/pattern-recognition-what-comes-next/.

Curio_Can. “What’s the next Number?” Puzzle Sphere, September 20, 2025. https://medium.com/puzzle-sphere/whats-the-next-number-e22903204bd8.

Hemanth. “How To Find The Next Number In The Sequence?” Street Science, May 25, 2023. https://medium.com/street-science/how-to-find-the-next-number-in-the-sequence-fc536a99f3f7. (Member Only Story)

Pierce, Rod. “Sequences.” Math Is Fun. Accessed February 26, 2026. https://www.mathsisfun.com/algebra/sequences-series.html.

Pierce, Rod. “Sequences – Finding A Rule.” Math Is Fun. Accessed February 26, 2026. https://www.mathsisfun.com/algebra/sequences-finding-rule.html.

Shah, Piyush N. “Find next Number in the Sequence 4,6,12,14,28,30 Calculator.” Accessed February 26, 2026. https://atozmath.com/NumberSeries.aspx.

Stapel, Elizabeth. “Finding Sequence Patterns with Common Differences.” Purplemath. Accessed February 26, 2026. https://www.purplemath.com/modules/nextnumb.htm.

Stapel, Elizabeth. “Finding the next Number: How to Think It Through.” Purplemath. Accessed February 26, 2026. https://www.purplemath.com/modules/nextnumb4.htm.

The Story of Mathematics. “Numbers – Number Sequences.” February 18, 2021. https://www.storyofmathematics.com/number-sequences/.

Thompson, Fletcher. “Can You Find the Next Number in the Sequence?” Puzzle Sphere, July 19, 2024. https://medium.com/puzzle-sphere/can-you-find-the-next-number-in-the-sequence-8795b153c48a. (Member Only Story)