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Number Sequences

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Practice your numerical reasoning skills with our number sequence questions. Each puzzle presents a series of numbers following a specific rule — your task is to identify the pattern and determine the next number in the sequence.

This format is commonly used in aptitude tests for employment and graduate programs. Our procedurally generated questions use only inferable patterns (no prior mathematical knowledge required), ensuring fair and unlimited practice opportunities.

All sequences are generated locally in your browser. Your progress and answers are stored only on your device — we never collect any personal data.

Key Patterns & Formats

Arithmetic Sequence

A sequence where each term differs from the previous by a constant amount (common difference).

aₙ = a₁ + (n-1)d, where d is the common difference

Geometric Sequence

A sequence where each term is found by multiplying the previous term by a constant (common ratio).

aₙ = a₁ × rⁿ⁻¹, where r is the common ratio

Fibonacci-Type Sequence

A sequence where each term is the sum of the two preceding terms.

aₙ = aₙ₋₁ + aₙ₋₂

Difference Method

A systematic technique for identifying sequence patterns by calculating differences between consecutive terms.

First differences: dₙ = aₙ₊₁ - aₙ

Second Differences

A pattern where the differences between consecutive terms form their own arithmetic sequence.

If second differences are constant = c, then aₙ = An² + Bn + C

Interleaved Sequences

Two or more independent sequences alternating within a single series.

Odd positions: a₁, a₃, a₅... follow Rule A; Even positions: a₂, a₄, a₆... follow Rule B

Compound Rules

Sequences that combine multiple operations (like add then multiply) to generate each term.

aₙ₊₁ = f(g(aₙ)), where f and g are different operations

Solving Strategies

  • Calculate the difference between consecutive terms
  • If all differences are the same, it's arithmetic
  • The pattern is 'add (or subtract) the same number each time'
  • Divide each term by the previous term
  • If all ratios are the same, it's geometric
  • The pattern is 'multiply by the same number each time'
  • Check if each term equals the sum of the two before it
  • Look for accelerating growth that's not purely multiplicative

Common Pitfalls

  • Confusing with geometric sequences (multiply vs add)
  • Miscounting when the common difference is negative
  • Forgetting that constant sequences (d=0) are also arithmetic
  • Confusing with arithmetic sequences (add vs multiply)
  • Forgetting that division is multiplication by a fraction
  • Missing negative ratios that cause alternating signs

Frequently Asked Questions

Number sequence questions assess your ability to identify patterns in series of numbers. You're given a sequence like '2, 4, 6, 8, ?' and must determine the rule (adding 2) to find the next number (10). These questions test logical reasoning and are widely used in employment aptitude tests.
Mostly no — but at harder difficulties you may encounter named sequences such as Fibonacci-like (each term = sum of the previous two), primes, squares, cubes, triangular numbers, factorials, and digit-sum patterns. Easy and most medium puzzles use only basic arithmetic progressions. If you don't know a named sequence, you can still solve most puzzles by looking for differences, ratios, or alternating steps between consecutive terms.
Patterns include: constant differences (+3, +3, +3), constant ratios (×2, ×2, ×2), increasing differences (+1, +2, +3), alternating operations (+5, -2, +5, -2), two interleaved sequences, and compound rules. All patterns can be discovered by examining the sequence — no prior knowledge required.

Sources & References

Arithmetic Sequence

Arithmetic Sequence
Find the next term: 2, 5, 8, 11, ?
  1. 1
    Calculate differences: 5-2=3, 8-5=3, 11-8=3Differences: 3, 3, 3
  2. 2
    All differences are the same: common difference = 3d = 3
  3. 3
    This is arithmetic. Add 3 to the last term11 + 3
  4. 4
    The next term is 1414
Refs:Math is Fun - Sequences and Series

Geometric Sequence

Geometric Sequence
aₙ = a₁ × rⁿ⁻¹, where r is the common ratio
2, 6, 18, 54, 162... (common ratio = ×3)
1000, 500, 250, 125... (common ratio = ×0.5 or ÷2)
1, -2, 4, -8, 16... (common ratio = ×(-2))
Find the next term: 2, 6, 18, 54, ?
  1. 1
    Calculate ratios: 6÷2=3, 18÷6=3, 54÷18=3Ratios: 3, 3, 3
  2. 2
    All ratios are the same: common ratio = 3r = 3
  3. 3
    This is geometric. Multiply the last term by 354 × 3
  4. 4
    The next term is 162162
Refs:Math is Fun - Sequences and Series

Fibonacci-Type Sequence

Fibonacci-Type Sequence
aₙ = aₙ₋₁ + aₙ₋₂
1, 1, 2, 3, 5, 8, 13, 21... (classic Fibonacci)
2, 5, 7, 12, 19, 31... (starts with 2 and 5)
1, 3, 4, 7, 11, 18... (Lucas numbers)
Find the next term: 1, 1, 2, 3, 5, 8, ?
  1. 1
    Check if each term = sum of previous two: 1+1=2 ✓1+1=2
  2. 2
    Continue checking: 1+2=3 ✓, 2+3=5 ✓, 3+5=8 ✓Pattern confirmed
  3. 3
    This is Fibonacci-type. Add the last two terms5 + 8
  4. 4
    The next term is 1313
Refs:Math is Fun - Sequences and Series

Difference Method

Difference Method
First differences: dₙ = aₙ₊₁ - aₙ
Sequence: 2, 5, 8, 11 → Differences: 3, 3, 3 (constant = arithmetic)
Sequence: 1, 4, 9, 16 → Differences: 3, 5, 7 → Second differences: 2, 2 (quadratic)
Sequence: 2, 6, 18, 54 → Differences: 4, 12, 36 (not constant, try ratios instead)
Find the pattern in: 1, 4, 9, 16, 25
  1. 1
    Calculate first differences: 4-1=3, 9-4=5, 16-9=7, 25-16=91st diff: 3, 5, 7, 9
  2. 2
    First differences not constant. Calculate second differences5-3=2, 7-5=2, 9-7=2
  3. 3
    Second differences are constant (2) - quadratic pattern2nd diff: 2, 2, 2
  4. 4
    These are perfect squares: 1², 2², 3², 4², 5²n² pattern
Refs:Math is Fun - Sequences and Series

Second Differences

Second Differences
If second differences are constant = c, then aₙ = An² + Bn + C
1, 4, 9, 16, 25... (squares) → 1st diff: 3, 5, 7, 9 → 2nd diff: 2, 2, 2
1, 3, 6, 10, 15... (triangular) → 1st diff: 2, 3, 4, 5 → 2nd diff: 1, 1, 1
2, 6, 12, 20, 30... → 1st diff: 4, 6, 8, 10 → 2nd diff: 2, 2, 2
Find the next term: 1, 3, 6, 10, 15, ?
  1. 1
    Calculate first differences: 3-1=2, 6-3=3, 10-6=4, 15-10=51st diff: 2, 3, 4, 5
  2. 2
    First differences increase by 1 each timeNot constant
  3. 3
    Second differences: 3-2=1, 4-3=1, 5-4=12nd diff: 1, 1, 1
  4. 4
    Next 1st diff = 6, so next term = 15 + 6 = 2121
Refs:Math is Fun - Sequences and Series

Interleaved Sequences

Interleaved Sequences
Odd positions: a₁, a₃, a₅... follow Rule A; Even positions: a₂, a₄, a₆... follow Rule B
Find the next term: 1, 10, 2, 20, 3, 30, ?
  1. 1
    Separate odd positions: 1, 2, 3 (positions 1, 3, 5)Odds: 1, 2, 3
  2. 2
    Separate even positions: 10, 20, 30 (positions 2, 4, 6)Evens: 10, 20, 30
  3. 3
    Odd positions: +1 pattern. Even positions: +10 patternTwo patterns
  4. 4
    Position 7 is odd. Next in odd sequence: 3+1 = 44
Refs:Math is Fun - Sequences and Series

Compound Rules

Compound Rules
aₙ₊₁ = f(g(aₙ)), where f and g are different operations
2, 5, 11, 23, 47... (×2 then +1 each step)
1, 3, 7, 15, 31... (×2 then +1, or 2ⁿ - 1)
3, 7, 15, 31... (×2 + 1 each step)
Find the next term: 2, 5, 11, 23, 47, ?
  1. 1
    Test differences: 3, 6, 12, 24 - not constant, but doubling!Differences double
  2. 2
    Test the rule: ×2 then +1 on each term2×2+1=5 ✓
  3. 3
    Continue: 5×2+1=11 ✓, 11×2+1=23 ✓, 23×2+1=47 ✓Rule confirmed
  4. 4
    Apply rule: 47×2+1 = 9595
Refs:Math is Fun - Sequences and Series