How to Solve Number Sequence Puzzles: Find the Next Number

Number sequence puzzles ask one simple-sounding question: what comes next? Behind that question is always a hidden rule, and once you find it, the next number falls out instantly. The good news is that the vast majority of number series follow a handful of common rules, and there's a reliable order to check them in. This guide gives you a step-by-step method to find the next number in any sequence, with worked examples for each rule type. If you want the broader picture across all pattern puzzles, start with how to solve pattern puzzles.

The one habit that solves most sequences

Before any technique, build this habit: write the differences between terms underneath the sequence. For 4, 7, 10, 13 you'd write the gaps 3, 3, 3. That single row of differences instantly reveals whether you're dealing with the simplest and most common rule of all. Most number series are cracked the moment you look at the gaps instead of the numbers.

Step 1: Check for a constant difference (arithmetic)

Subtract each term from the next. If the gap is the same every time, it's an arithmetic sequence, and you find the next number by adding that gap once more.

• 5, 9, 13, 17, ... gaps are all +4, so the next number is 21.
• 20, 17, 14, 11, ... gaps are all −3, so the next number is 8.

Arithmetic sequences are the most common, so this is always your first check.

Step 2: Check for a constant ratio (geometric)

If the differences aren't constant, divide each term by the one before it. If that ratio is the same every time, it's a geometric sequence, and you multiply to get the next number.

• 3, 6, 12, 24, ... each term doubles (×2), so the next is 48.
• 80, 40, 20, 10, ... each term halves (×0.5), so the next is 5.

Arithmetic adds, geometric multiplies. Between these two rules you'll solve the large majority of easy and medium sequences.

Step 3: Look at the second differences

When the gaps aren't constant, look at how the gaps themselves change. Write a second row of differences underneath the first.

Take 2, 6, 12, 20, 30. The first differences are 4, 6, 8, 10. Those aren't constant, but their differences are: 2, 2, 2. A constant second difference means the sequence follows a quadratic (position-squared) rule. The next first-difference is 12, so the next number is 30 + 12 = 42.

Whenever the second row of differences is constant, you're looking at a position-based formula like n² or n(n+1), which brings us to the next check.

Step 4: Check position-based rules

Sometimes the rule depends on the term's position rather than the previous term. Compare each term to its position number (1st, 2nd, 3rd...):

• 1, 4, 9, 16, 25, ... these are 1², 2², 3², 4², 5², the square numbers. Next is 6² = 36.
• 1, 3, 6, 10, 15, ... these are the triangular numbers, n(n+1)/2. Next is 21.
• 2, 6, 12, 20, 30, ... these are n × (n+1). Next is 6 × 7 = 42 (the same sequence from step 3, seen another way).

If a sequence grows faster than a constant difference but isn't cleanly geometric, suspect a position-based formula.

Step 5: Split alternating and interleaved sequences

Some "number series" are really two sequences woven together. When no single rule fits, split the terms into odd positions and even positions and look at each half on its own.

• 1, 10, 2, 20, 3, 30, ... odd positions are 1, 2, 3 (next 4) and even positions are 10, 20, 30 (next 40). The next number is 4.
• 3, 4, 6, 8, 12, 16, ... odd positions are 3, 6, 12 (×2) and even positions are 4, 8, 16 (×2). The next number is 24.

This "split it in two" move catches most patterns that look random at first glance, and it's the key technique on our medium pattern puzzles.

Step 6: Recognize the famous sequences

A few sequences appear so often that you should learn them on sight:

• Fibonacci: each term is the sum of the previous two. 1, 1, 2, 3, 5, 8, 13, ... next is 21.
• Primes: 2, 3, 5, 7, 11, 13, ... next is 17.
• Powers of 2: 1, 2, 4, 8, 16, 32, ...
• Squares and cubes: 1, 4, 9, 16 and 1, 8, 27, 64.

If a sequence resists differences and ratios, check whether it's one of these. There's a fuller reference in types of number patterns.

A worked example, start to finish

Find the next number in 5, 11, 23, 47.

• Differences: 6, 12, 24. Not constant, so it's not arithmetic.
• Ratios: 11/5 isn't clean, so it's not simply geometric on the terms.
• Look at the differences again: 6, 12, 24 double each time. So the rule is "multiply the gap by 2," or equivalently each term is (previous × 2 + 1). Check: 5×2+1 = 11, 11×2+1 = 23, 23×2+1 = 47. The next number is 47×2+1 = 95.

Notice how working the differences pointed straight to the rule. That's the method in action.

When to stop and use a hint

If a sequence resists every check above for about 30 seconds, it's usually a composite rule (two operations combined) found on expert puzzles. At that point a hint that reveals the rule teaches you a pattern you'll recognize next time, which is far more useful than guessing. Don't grind, learn the rule and move on.

Practice makes it instant

The checks here become automatic with repetition. Soon you'll glance at a sequence and feel whether it's arithmetic, geometric, or interleaved before you've consciously done the math. Start on our easy pattern puzzles to drill differences and ratios, then climb toward the expert sequences where composite rules await.

Frequently asked questions

How do you find the next number in a sequence?

Write the differences between terms. If the difference is constant, add it again (arithmetic). If not, check the ratio between terms (geometric). If neither is constant, look at the second differences, compare terms to their position numbers, or split alternating sequences into two halves.

What is the rule for a number sequence?

The rule is the operation that turns each term into the next, or that maps a position to its value. Common rules are add a constant (arithmetic), multiply by a constant (geometric), square the position, or sum the previous two terms (Fibonacci). A correct rule must explain every term, not just some.

How do you solve number series questions fast?

Check the simplest rules first: constant difference, then constant ratio. Together those solve most series. Build the habit of writing the gaps under the numbers, because that instantly reveals arithmetic sequences and points you toward the right rule for the rest.

What are the most common number sequences?

Arithmetic (constant difference), geometric (constant ratio), square numbers, triangular numbers, the Fibonacci sequence, and prime numbers. Recognizing these on sight solves a large share of number sequence puzzles immediately.