Number Sequence Calculator — Free Online Sequence Analyzer

How to Use the Number Sequence Calculator

1. Select the mode: Choose from four options using the dropdown. "Analyze Sequence" identifies patterns in your data. "Generate Arithmetic," "Generate Geometric," and "Generate Fibonacci" create new sequences from parameters you specify.
2. For sequence analysis: Enter at least 3 numbers separated by commas in the text field. For example, "2, 5, 8, 11, 14" for an arithmetic sequence or "3, 6, 12, 24" for a geometric one. The more terms you provide, the more confident the pattern detection.
3. For sequence generation: Enter the first term, the common difference (arithmetic) or common ratio (geometric), and the desired number of terms. For Fibonacci, just specify the term count.
4. Read the analysis: For analyzed sequences, the calculator identifies the type (arithmetic, geometric, Fibonacci-like, or unknown), displays the formula, describes the pattern, and predicts the next 5 terms.
5. Check the sum: For generated sequences, the calculator computes the sum of all terms, which is useful for series calculations in mathematics, physics, and financial modeling.

The calculator handles both integers and decimals, and can detect patterns in sequences with negative terms or fractional ratios. Results update instantly as you modify the input.

Sequence Formulas

Variables Explained

• a(n): The nth term of the sequence, where n starts at 1 (or sometimes 0 for Fibonacci).
• a(1): The first term of the sequence, which serves as the starting point for both formulas.
• d: The common difference in an arithmetic sequence — the constant value added to get each next term.
• r: The common ratio in a geometric sequence — the constant multiplier applied to get each next term.
• S(n): The sum of the first n terms of the sequence (the partial sum or series).

Step-by-Step Example

Analyze the sequence: 5, 15, 45, 135, 405

1. Check for arithmetic pattern: differences are 10, 30, 90, 270 (not constant). Not arithmetic.
2. Check for geometric pattern: ratios are 15/5=3, 45/15=3, 135/45=3, 405/135=3 (constant!). Geometric with r=3.
3. Formula: a(n) = 5 x 3^(n-1)
4. Predict next terms: 405 x 3 = 1,215; 1,215 x 3 = 3,645; 3,645 x 3 = 10,935

The sequence is geometric with first term 5 and common ratio 3. Each term is exactly 3 times the previous one. The sum of the 5 given terms is 5 + 15 + 45 + 135 + 405 = 605, which can be verified with the formula: 5 x (1 - 3^5) / (1 - 3) = 5 x (-242) / (-2) = 605.

Practical Examples

Example 1: Amanda's Savings Plan

Amanda saves money each month, increasing her deposit by $50 each time. She starts with $200 in January. She wants to know her deposit in December and total savings for the year.

• Arithmetic sequence: a(1) = 200, d = 50
• December (12th month): a(12) = 200 + 50(12-1) = 200 + 550 = $750
• Total: S(12) = 12/2 x (200 + 750) = 6 x 950 = $5,700

Amanda will deposit $750 in December and save $5,700 total over the year. Arithmetic sequences model any situation with constant linear growth, from salary increases to equally spaced measurement points. For compound growth scenarios, try our compound interest calculator.

Example 2: Brandon's Bacteria Culture

Brandon is studying bacteria growth in biology class. His culture doubles every hour, starting with 500 bacteria. He needs to predict the population after 8 hours.

• Geometric sequence: a(1) = 500, r = 2
• After 8 hours: a(9) = 500 x 2^8 = 500 x 256 = 128,000
• Sequence: 500, 1000, 2000, 4000, 8000, 16000, 32000, 64000, 128000

The population reaches 128,000 after 8 hours — the power of exponential growth. Geometric sequences model biological growth, radioactive decay, compound interest, and any process with a constant growth rate. Note that doubling time is constant in geometric growth, which is why it quickly produces dramatically large numbers.

Example 3: Linda's Pattern Recognition Test

Linda encounters this sequence on a math competition: 1, 1, 2, 3, 5, 8, 13. She needs to identify the pattern and find the next three terms.

• Check differences: 0, 1, 1, 2, 3, 5 (not constant — not arithmetic)
• Check ratios: 1, 2, 1.5, 1.67, 1.6, 1.625 (not constant — not geometric)
• Check Fibonacci: 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13 (each term = sum of previous two)
• Next three: 8+13=21, 13+21=34, 21+34=55

Linda identifies a Fibonacci sequence. The next terms are 21, 34, and 55. Fibonacci patterns appear throughout nature — in the spiral arrangement of sunflower seeds, the branching of trees, and the proportions of the human body. The ratio of consecutive terms approaches the golden ratio (phi = 1.618...), which appears in art, architecture, and design.

Sequence Types Reference Table

Tips and Complete Guide

Pattern Detection Strategy

When analyzing a sequence, check for patterns in this order: (1) compute differences between consecutive terms — if constant, it is arithmetic; (2) compute ratios between consecutive terms — if constant, it is geometric; (3) check if each term equals the sum of the two before it — if yes, it is Fibonacci-like. If none of these work, compute second differences (differences of differences) — if those are constant, the sequence follows a quadratic pattern. Our calculator handles the first three types automatically.

Real-World Applications

Arithmetic sequences model linear phenomena: constant salary increases ($2,000/year raise), uniform depreciation, evenly spaced measurements, and simple interest. Geometric sequences model exponential phenomena: compound interest, population growth, radioactive decay, and signal attenuation. Fibonacci sequences appear in biological growth patterns, algorithm analysis (Fibonacci heaps), and financial market analysis (Fibonacci retracements). Understanding which model applies to a real-world situation is a key skill in applied mathematics.

Convergence and Divergence

Arithmetic sequences always diverge (grow without bound) unless the common difference is 0. Geometric sequences converge to 0 when |r| is less than 1 and diverge when |r| is greater than 1. The infinite geometric series 1 + r + r^2 + r^3 + ... converges to 1/(1-r) only when |r| is less than 1. This convergence property is fundamental in calculus, signal processing, and financial mathematics (present value of perpetuities).

Sequences in Computer Science

Sequences are fundamental to algorithm analysis. The time complexity of merge sort follows the recurrence T(n) = 2T(n/2) + n (related to geometric growth). Fibonacci numbers directly relate to the worst-case performance of the Euclidean algorithm. Binary search halves the search space each step (geometric sequence with r=1/2). Dynamic programming often involves computing sequences where each term depends on previous ones, much like Fibonacci computation.

Common Mistakes to Avoid

• Too few terms for reliable detection: With only 2 terms, any pattern could fit. Provide at least 3-4 terms for arithmetic/geometric detection. More terms increase confidence, especially for distinguishing quadratic from arithmetic patterns.
• Confusing arithmetic and geometric growth: Arithmetic means constant addition (linear), geometric means constant multiplication (exponential). A sequence growing by 10% each term is geometric (r=1.1), not arithmetic.
• Rounding errors in ratio detection: When ratios are approximately but not exactly constant (e.g., 2.001, 1.999, 2.002), the sequence may be approximately geometric. Small floating-point differences are normal in real-world data.
• Assuming all Fibonacci-like sequences start with 0, 1: Any sequence where each term is the sum of the two preceding terms is Fibonacci-like. Lucas numbers (2, 1, 3, 4, 7, 11...) follow the same rule with different starting values.
• Extrapolating too far: Even correctly identified patterns may break down in real-world contexts. Exponential growth cannot continue forever in physical systems, and predictive accuracy decreases with distance from the known data.

Frequently Asked Questions