Number Sequence Calculator
Find the nth term of arithmetic, geometric, and other common sequences.
What Is the Number Sequence Calculator?
The Number Sequence Calculator identifies and extends arithmetic, geometric, Fibonacci, and other mathematical sequences. Enter the known terms of a sequence and the calculator determines the type, finds the common difference or ratio, and generates subsequent terms with the general formula.
Formula
How to Use
Enter the first several terms of your sequence separated by commas. The calculator identifies whether the sequence is arithmetic (constant difference), geometric (constant ratio), or another recognizable pattern. It then generates the next N terms and displays the closed-form formula.
Example Calculation
Arithmetic: 3, 7, 11, 15 → d=4, aₙ = 3+(n−1)×4 = 4n−1. Term 10 = 39. Geometric: 2, 6, 18, 54 → r=3, aₙ = 2×3ⁿ⁻¹. Term 8 = 2×3⁷ = 4374. Fibonacci: 1, 1, 2, 3, 5, 8 → Fₙ = Fₙ₋₁+Fₙ₋₂.
Understanding Number Sequence
Mathematical sequences are ordered lists of numbers following a specific rule or pattern. The most common types are arithmetic sequences (constant difference between terms) and geometric sequences (constant ratio between terms). Recognizing and formulating these patterns is a foundational skill in algebra, calculus, and discrete mathematics.
Arithmetic sequences model linear growth or decay: equal amounts added or removed each period. Examples include monthly savings deposits, evenly spaced data points, and arithmetic progressions in music theory. Geometric sequences model exponential growth or decay: compound interest, population growth, radioactive decay, and drug concentration in the body.
Beyond arithmetic and geometric sequences, the Fibonacci sequence (each term is the sum of the two preceding ones: 1, 1, 2, 3, 5, 8, 13, ...) appears in plant spirals, shell proportions, and financial market analysis. The Tribonacci, Lucas, and other recurrence sequences have analogous closed-form expressions and widespread mathematical importance.
Frequently Asked Questions
How do I identify an arithmetic sequence?
An arithmetic sequence has a constant difference between consecutive terms: aₙ₊₁ − aₙ = d (constant). Examples: 2, 5, 8, 11 (d=3); 10, 7, 4, 1 (d=−3).
How do I identify a geometric sequence?
A geometric sequence has a constant ratio between consecutive terms: aₙ₊₁ / aₙ = r (constant). Examples: 2, 6, 18, 54 (r=3); 100, 10, 1, 0.1 (r=0.1).
What is the sum formula for arithmetic sequences?
Sum of n terms of an arithmetic sequence: Sₙ = n(a₁ + aₙ)/2 = n[2a₁ + (n−1)d]/2.
What is the sum of a geometric series?
Finite sum: Sₙ = a₁(1−rⁿ)/(1−r) for r≠1. Infinite sum (|r|<1): S∞ = a₁/(1−r).
Is this calculator free?
Yes, completely free with no sign-up required.
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