Three Number Sequences to Test Your Pattern Spotting

Number sequence puzzles ask you to find the rule connecting a short list of numbers and then apply that same rule one more time. The trick is rarely complicated arithmetic; it's noticing which operation stays constant across every step. Below are three sequences, each checked by generating the actual rule and confirming it matches every given number before predicting the next one.

Sequence 1

What comes next?

1, 2, 6, 24, 120, ?

Sequence 2

What comes next?

12, 20, 30, 42, 56, ?

Sequence 3

What comes next?

29, 31, 37, 47, 61, 79, ?

How to spot each pattern

For Sequence 1, check the ratio between consecutive terms rather than the difference. 2 divided by 1 is 2. 6 divided by 2 is 3. 24 divided by 6 is 4. 120 divided by 24 is 5. The multiplier itself is increasing by one each time (×2, then ×3, then ×4, then ×5), so the next multiplier is ×6.

For Sequence 2, differences between terms are 8, 10, 12, 14, which grow by 2 each time, a common sign that the sequence follows a formula based on position. Checking position: the first term (12) equals 3×4, the second (20) equals 4×5, the third (30) equals 5×6. Each term is n times (n+1) for consecutive values of n, so the pattern is the product of two consecutive integers.

For Sequence 3, look at the gaps between terms: 31−29=2, 37−31=6, 47−37=10, 61−47=14, 79−61=18. These gaps themselves increase by 4 each time (2, 6, 10, 14, 18), so the next gap should be 22.

Checking each answer

Generating each sequence from its rule reproduces every given number exactly, not just the last one or two, so each rule is confirmed before it is used to predict the missing number.

Answers

Sequence 1: 720. The sequence is 1!, 2!, 3!, 4!, 5!, so the next term is 6! = 720.

Sequence 2: 72. Each term is n×(n+1) for increasing n (3×4, 4×5, 5×6, 6×7, 7×8), so the next term is 8×9 = 72.

Sequence 3: 101. The gaps between terms grow by 4 each time (2, 6, 10, 14, 18), so the next gap is 22, and 79 + 22 = 101.

A general method for any new sequence you see

When you hit a sequence that doesn't immediately look familiar, work through these checks in order, since each one is quick to test and rules out a whole category of pattern:

1. Check the simple difference between consecutive terms first. If the difference is constant, you have a straightforward arithmetic sequence, and the next term is just the last term plus that constant difference.
2. If the difference isn't constant, check whether the difference itself changes by a constant amount, the way Sequence 3 above does (gaps of 2, 6, 10, 14, 18, each one 4 more than the last). This pattern, sometimes called a second-order arithmetic sequence, is common in puzzle sets specifically because it looks irregular at first glance but resolves cleanly once you look one level deeper.
3. Check the ratio between consecutive terms, especially if the numbers are growing quickly. A constant ratio means a simple geometric sequence (multiply by the same number every time). A ratio that itself increases by a fixed amount, as in Sequence 1 above (×2, ×3, ×4, ×5), points toward a factorial-style pattern.
4. Check whether terms match a position-based formula, such as n squared, n times (n+1), or triangular numbers (1, 3, 6, 10, 15...). Lining up each term with its position in the sequence (first, second, third...) and testing a few common formulas is often faster than staring at the raw numbers alone.
5. If none of the above work on their own, look for two interleaved sequences. Some puzzle sets alternate between two separate simple patterns at odd and even positions; splitting the sequence into two shorter lists sometimes reveals a pattern that was invisible in the combined list.

This order matters because it goes from fastest-to-check to slowest-to-check. Most number sequence puzzles you'll encounter, including all three above, resolve at step 1, 2, or 3, so it's worth exhausting those before trying anything more elaborate.

Why checking every term matters, not just the last two

It's tempting to look at only the last two numbers in a sequence, find a relationship between them, and assume that relationship explains the whole list. This is risky because a rule that happens to fit two numbers by coincidence is common, especially with small numbers, but a rule that fits five or six numbers in a row is much less likely to be a coincidence. This is exactly why the check above checks the rule against the entire given sequence, 1 through 120 for Sequence 1, all five terms for Sequence 2, and all six terms for Sequence 3, rather than just confirming the last step before predicting the next number.

Key takeaways

• When differences between terms don't look obviously constant, check the ratio between terms instead; a growing multiplier is a common pattern.
• If differences grow by a constant amount, check whether the terms match a simple formula like n×(n+1), which produces exactly that kind of growing-gap pattern.
• Work through checks in order of speed: constant difference, then growing difference, then constant ratio, then growing ratio, then position-based formulas, then interleaved sequences.
• Always verify a suspected rule against every given number in the sequence, not just the last step, before trusting your predicted answer.
• Writing the rule as actual code and running it is the most reliable way to catch a pattern that only "mostly" fits.