Number Sequence Calculator

Understanding number sequences for students and professionals

Number sequences are ordered lists of values that follow a rule. They power forecasting, optimization, signal processing, financial modeling, and research methods. Recognizing patterns and applying formulas lets you compute terms directly, summarize behavior with sums, and validate models with clear assumptions.

Essential sequence types

• Arithmetic: A constant difference d between terms (e.g., 2, 5, 8…). Nth term: aₙ = a₁ + (n − 1)d. Sum: Sₙ = n/2 · [2a₁ + (n − 1)d].
• Geometric: A constant ratio r between terms (e.g., 3, 6, 12…). Nth term: aₙ = a₁ · r^(n − 1). Sum (r ≠ 1): Sₙ = a₁ · (1 − rⁿ)/(1 − r).
• Fibonacci: Each term is the sum of the previous two with seeds a₁, a₂ (e.g., 0, 1, 1, 2…). Useful in combinatorics, growth models, and algorithms.
• Linear recurrence: aₙ = p·aₙ₋₁ + q captures growth/decay and shift processes in control, finance, and time series.

Detecting patterns

Constant differences point to arithmetic; constant ratios suggest geometric; sums of previous terms indicate Fibonacci-like relations. When a list doesn’t fit neatly, treat it as a custom recurrence or compute terms numerically to preserve integrity.

Best practices

• State formulas: Include nth term and sum expressions for transparency.
• Avoid rounding drift: Use adequate precision when ratios or differences are fractional.
• Check domains: Geometric sums require r ≠ 1; very large Fibonacci N grows quickly.
• Document seeds: Seeds define behavior for Fibonacci and recurrences; record them in outputs.

Applications

• Finance: Regular deposits (arithmetic) and compound growth (geometric).
• Engineering: Filters and decays modeled by linear recurrences.
• Data science: Feature engineering with sequence patterns and cumulative sums.

Frequently asked questions

HOW TO generate an arithmetic sequence and its sum?

Steps: Choose Arithmetic, set a1 and d, enter N and n for nth term, enable Include partial sum if needed, then Calculate.

HOW TO compute a geometric sequence and partial sum?

Select Geometric, provide a1 and r, set N and n, and Calculate. For r ≠ 1, the sum uses a₁ · (1 − rⁿ)/(1 − r).

HOW TO create a Fibonacci sequence with custom seeds?

Choose Fibonacci, enter seeds a1 and a2, set N and n, and Calculate to get the list and the nth term.

HOW TO build a linear recurrence sequence?

Select Custom recurrence, set seed a1 and coefficients p, q for aₙ = p·aₙ₋₁ + q, choose N and n, then Calculate.

HOW TO analyze a pasted list of terms?

Choose Analyze given terms, paste your list, and Calculate. The tool checks differences, ratios, predicts the next term, and labels the pattern.

HOW TO format results for reports?

Use Output format = Plain, Compact, CSV, or LaTeX to copy directly into documents, spreadsheets, or papers.

HOW TO embed this calculator on my website?

Copy this single-file HTML page into your project, link it in navigation, or embed via iframe with a descriptive title.