Number Sequence Calculator
Identify arithmetic, geometric, and Fibonacci sequences. Find common difference, common ratio, nth-term formula, and compute next terms.
Number Sequences
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The Formula
A number sequence is an ordered list of numbers that follow a pattern. An arithmetic sequence has a constant difference between consecutive terms. A geometric sequence has a constant ratio. A Fibonacci sequence adds the two previous terms to produce the next. Identifying the pattern allows you to predict future terms and find a closed-form formula.
Variable Definitions
First Term
The starting value of the sequence. The foundation for building all subsequent terms.
Common Difference
Constant difference between consecutive terms in an arithmetic sequence. Can be positive, negative, or zero.
Common Ratio
Constant ratio between consecutive terms in a geometric sequence. Can be any non-zero number including fractions and decimals.
Term Position
The index of the term in the sequence, starting at 1 for the first term.
How to Use This Calculator
- 1
Enter at least 3 comma-separated numbers from your sequence (more terms improve pattern detection accuracy).
- 2
Choose how many additional terms to compute (up to 20).
- 3
The calculator identifies the pattern type — arithmetic, geometric, or Fibonacci — and displays the nth-term formula.
- 4
Review the step-by-step reasoning for how the pattern was detected, including the common difference or ratio.
- 5
For unknown patterns, the calculator reports that no simple pattern was found — your sequence may follow a higher-order rule.
Sequences follow patterns: arithmetic adds a constant difference, geometric multiplies by a constant ratio
Understanding the Concept
Sequence recognition is a fundamental skill in mathematics and computer science. Arithmetic sequences describe linear growth (constant addition) — like saving a fixed amount each month. Geometric sequences describe exponential growth (constant multiplication) — like compound interest or population growth. Fibonacci sequences appear throughout nature — from spiral shells and pinecones to branching trees and leaf arrangements (phyllotaxis). The nth-term formula (or closed-form formula) lets you compute any term directly without enumerating all preceding terms, which is essential for large indices. For example, the 100th term of an arithmetic sequence with first term 1 and difference 3 is a(100) = 1 + 3(99) = 298 — much faster than adding 3 ninety-nine times. Similarly, the sum of the first n terms of an arithmetic sequence is S(n) = n(a₁ + aₙ)/2, and for a geometric sequence it is S(n) = a₁(rⁿ − 1)/(r − 1). These summation formulas appear in calculus (Riemann sums), finance (annuity calculations), and computer science (algorithm analysis).
Frequently Asked Questions
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