Number Sequence Calculator Identify, extend, and analyze arithmetic, geometric, Fibonacci, and quadratic sequences — with...

Number Sequence Calculator

Identify, extend, and analyze arithmetic, geometric, Fibonacci, and quadratic sequences — with pattern detection and real-world insights.

Detect Pattern

Arithmetic

Geometric

Fibonacci

Quadratic

Sequence Terms (comma-separated)

Next Terms

First Term (a₁)

Common Difference (d)

n (term #)

First Term (a₁)

Common Ratio (r)

n (term #)

Type Classic Fibonacci Lucas Numbers Tribonacci

n (term #)

a (n² coefficient)

b (n coefficient)

c (constant)

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n-th Term

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Next Terms

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Sum of First n

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Pattern

Sequence Plot & Differences

Sequence Formulas

Arithmetic**:

$$ a_n = a_1 + (n-1)d, \quad S_n = \frac{n}{2}(2a_1 + (n-1)d) $$

Geometric**:

$$ a_n = a_1 r^{n-1}, \quad S_n = a_1 \frac{1 - r^n}{1 - r} \ (r \ne 1) $$

Fibonacci**:

$$ F_1 = 1, F_2 = 1, \ F_n = F_{n-1} + F_{n-2} $$

Quadratic** (e.g., 2, 5, 10, 17…): First diff: 3, 5, 7 → Second diff: 2, 2 → **aₙ = n² + 1**

✅ Pro Tip**: For unknown sequences, compute **first & second differences** — constant 1st = arithmetic; constant 2nd = quadratic; constant ratio = geometric.

Pattern Detection Red Flags

⚠️ Avoid these errors:

• Assuming linearity** — 1, 2, 4, 8 looks linear early but is geometric
• Ignoring signs** — 5, 2, −1, −4 is arithmetic (d = −3), not “random”
• Overfitting noise** — Real data has error; don’t force 5th-degree polynomial on 5 points
• Missing recursion** — 1, 1, 2, 3, 5 is Fibonacci, not quadratic

✅ Real-World Uses**:

• Finance**: $1000 × 1.05ⁿ = compound growth
• CS**: Merge sort cost = n·log₂n (not arithmetic/geometric!)
• Biology**: Rabbit pairs → Fibonacci; sunflower seeds → Fibonacci spirals

2025 Sequence Benchmarks

Sequence	First 5 Terms	Type	10th Term
Arithmetic (d=3)	2, 5, 8, 11, 14	aₙ = 3n−1	29
Geometric (r=2)	3, 6, 12, 24, 48	aₙ = 3·2ⁿ⁻¹	1536
Fibonacci	1, 1, 2, 3, 5	Fₙ = Fₙ₋₁ + Fₙ₋₂	55
Quadratic	2, 5, 10, 17, 26	n² + 1	101

📉 Difference Table for 2, 5, 10, 17, 26**:

n      aₙ     Δ¹     Δ²
1      2
              3
2      5             2
              5
3     10             2
              7
4     17             2
              9
5     26
        

→ **Δ² = 2 constant ⇒ quadratic**

How to Use This Calculator

➡️ Detect Pattern

“2, 5, 8, 11” → **Arithmetic, d=3**, next: 14, 17, 20

➡️ Arithmetic

“a₁=3, d=4, n=10” → **a₁₀ = 39**, sum = 210

➡️ Geometric

“a₁=2, r=3, n=6” → **a₆ = 486**, sum = 728

➡️ Quadratic

“a=1, b=1, c=0” → **aₙ = n² + n**, terms: 2, 6, 12, 20…

Note: All sequences support decimals and negatives. Fibonacci limited to n ≤ 50 for performance.