Sequence Analyzer | Detect Pattern, nth Term & Generate Terms
Enter any numerical sequence and automatically detect whether it is arithmetic, geometric, quadratic, cubic, Fibonacci-type, or power-based. Finds the closed-form nth term formula, builds a complete difference table at every order, generates the next N terms, and computes partial sums.
What Is the Sequence Analyzer | Detect Pattern, nth Term & Generate Terms?
A sequence is an ordered list of numbers. The key question is: what is the rule? The method of finite differences tests progressively higher-order differences (Δ¹, Δ², Δ³) to identify polynomial sequences — if the k-th differences are constant, the sequence is a k-th degree polynomial. Geometric and Fibonacci patterns require their own checks. Finding the pattern gives a closed-form nth-term formula that can generate any term without computing all previous terms.
Formula
Arithmetic: aₙ = a₀ + d·n (Δ¹ constant)
Geometric: aₙ = a₀ · rⁿ (ratios constant)
Quadratic: aₙ = An² + Bn + C (Δ² constant)
Fibonacci: aₙ = aₙ₋₁ + aₙ₋₂
Power: aₙ = c · nᵏ (k = 2, 3, 4)
How to Use
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Enter your numerical sequence separated by spaces or commas
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Or click a preset: Arithmetic, Geometric, Squares, Fibonacci, Cubes, Primes
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Click Analyze Sequence
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Read the detected type, formula, and confidence score
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View next 5 predicted terms and partial sum of given terms
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Open the difference table to see Δ⁰, Δ¹, Δ² levels
Enter your sequence as space- or comma-separated numbers. Click Analyze Sequence. The analyzer identifies the type, returns the nth-term formula, lists the next 5 predicted terms, and displays the complete difference table. Use presets for quick demonstrations.
Example Calculation
Example: 1, 4, 9, 16, 25, 36 (perfect squares)
Δ¹: 3, 5, 7, 9, 11 (first differences)
Δ²: 2, 2, 2, 2 (constant!) → Quadratic
Formula: aₙ = n² (where n starts at 1)
Next terms: 49, 64, 81, 100, 121
Frequently Asked Questions
How does the difference table work?
The zeroth differences (Δ⁰) are the original sequence terms. The first differences (Δ¹) are consecutive differences: a₁−a₀, a₂−a₁, …. The second differences (Δ²) are differences of Δ¹, and so on. If the k-th row is constant, the sequence follows a k-th degree polynomial. This is Newton's forward difference interpolation method.
What if the sequence is not recognized?
The analyzer reports "Unknown / Complex" when the sequence does not match any built-in patterns (arithmetic, geometric, quadratic, cubic, Fibonacci, or power n^k). This can happen with prime numbers, logarithmic sequences, or sequences defined by more complex rules. More terms often help the analyzer identify the pattern.
How accurate are the predicted next terms?
Predictions assume the detected pattern continues exactly. For arithmetic and geometric sequences the formula is exact. For quadratic and cubic sequences the polynomial formula is exact if the sequence is truly polynomial. Fibonacci-type sequences use the recurrence directly. Predictions break down if the true sequence has an exception or changes pattern.
Why is the primes sequence labeled as unknown?
There is no simple closed-form polynomial or exponential formula for the sequence of prime numbers. The nth prime grows roughly as n·ln(n) by the Prime Number Theorem, but this is an approximation, not an exact formula. The analyzer correctly identifies primes as "Unknown / Complex".
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