Number Sequence Puzzles With Answers (And How to Solve Any of Them)

Short answer: Solve number sequences by checking, in order: differences between terms, then differences of those differences, then ratios, then whether terms combine (like Fibonacci), then whether it is two sequences interleaved. That checklist solves the large majority of sequence questions in aptitude tests. Practise on Number Sequence.

The method

Work through these in order and stop when something fits.

Step 1: First differences

Subtract each term from the next. If the result is constant, it is arithmetic.

3, 7, 11, 15, ? → differences 4, 4, 4 → next is 19.

Step 2: Second differences

If first differences are not constant, take differences of those.

2, 5, 10, 17, 26, ? → first differences 3, 5, 7, 9 → second differences 2, 2, 2. Next first difference is 11, so the answer is 37.

These are usually squares in disguise — this one is n² + 1.

Step 3: Ratios

Divide each term by the previous. Constant result means geometric.

3, 6, 12, 24, ? → ratio 2 → 48.

Step 4: Combination of previous terms

Check whether each term comes from the ones before it.

1, 1, 2, 3, 5, 8, ? → each is the sum of the two before → 13. Fibonacci.

Variants exist: sum of the previous three, previous term doubled plus one, and so on.

Step 5: Interleaved sequences

If nothing else fits, try reading alternate terms as two separate sequences.

2, 100, 4, 90, 6, 80, ? → odd positions 2, 4, 6 (+2); even positions 100, 90, 80 (−10). Next is position 7, odd, so 8.

This is the one most people miss, and test-setters know it.

Step 6: Operation patterns

Sometimes the operation itself changes in a pattern.

2, 3, 6, 18, 108, ? → each term is the product of the two before → 1944.

Worked examples

Easy

5, 10, 20, 40, ?
Ratio 2. Answer: 80.

1, 4, 9, 16, ?
Perfect squares. Answer: 25.

100, 92, 84, 76, ?
Difference −8. Answer: 68.

Medium

1, 2, 6, 24, 120, ?
Multiply by 2, 3, 4, 5 → next multiply by 6. Answer: 720. Factorials.

7, 10, 16, 28, ?
Differences 3, 6, 12 — doubling. Next difference 24. Answer: 52.

2, 3, 5, 7, 11, 13, ?
Prime numbers. Answer: 17.

4, 9, 16, 25, ?
Squares of 2, 3, 4, 5. Answer: 36.

Harder

1, 3, 7, 15, 31, ?
Each term is double the previous plus one. Answer: 63. Also 2ⁿ − 1.

8, 6, 9, 23, 87, ?
Multiply by 1 and subtract 2; by 2 and subtract 3; by 3 and subtract 4; by 4 and subtract 5. Next: 87 × 5 − 6 = 429.

3, 8, 15, 24, 35, ?
Differences 5, 7, 9, 11 → next 13. Answer: 48. Also n² − 1.

1, 11, 21, 1211, 111221, ?
The look-and-say sequence — each term describes the one before. "111221" reads as three 1s, two 2s, one 1. Answer: 312211.

Patterns worth memorising on sight

• Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
• Cubes: 1, 8, 27, 64, 125, 216
• Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
• Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34
• Factorials: 1, 2, 6, 24, 120, 720
• Powers of 2: 2, 4, 8, 16, 32, 64, 128
• Triangular: 1, 3, 6, 10, 15, 21

Recognising these instantly is most of the speed advantage in a timed aptitude test. If a sequence looks close to one of these, check whether it is that pattern shifted by a constant — n² + 1 and n² − 1 appear constantly.

Exam tactics

• Write the differences down. Doing it mentally causes errors under time pressure.
• Check your answer forwards. Apply your rule from the first term and confirm every term matches, not just the last.
• Watch for sign changes. Alternating positive and negative usually means multiplication by a negative number.
• Move on quickly. If nothing fits in 60 seconds, mark it and return. Sequence questions have a nasty habit of becoming obvious on a second look.
• Use the options. In multiple choice, testing the given answers backwards is often faster than deriving the rule.

Practise timed sequences on Number Sequence, and try Math Sprint for the mental arithmetic speed these questions assume.

Related reading: free aptitude test practice and mental maths tricks.