Identify and compute arithmetic, geometric, Fibonacci, triangular, and other number sequences. Find the nth term, sum, and pattern analysis for common sequences.
The Number Sequence Calculator generates terms, finds the nth term, and computes partial sums for common mathematical sequences. It supports arithmetic sequences, geometric sequences, Fibonacci-type recurrences, triangular numbers, square numbers, prime numbers, and other familiar patterns. It is a quick way to tell whether a sequence is linear, exponential, or recursive before you write the formula yourself. That makes it easier to spot the pattern behind a table of values. It is also a quick check before you commit a guessed rule to code or homework.
Sequences appear throughout mathematics, programming, and modeling. Arithmetic sequences describe steady step-by-step change, geometric sequences describe exponential growth or decay, and recursive sequences show how each term depends on what came before.
Choose a sequence type, enter the starting values or parameters, and the calculator will return the terms, direct formula, and cumulative sum behavior in one place. That makes it useful both for homework checks and for quickly testing pattern logic in code.
It saves time when you need to identify a pattern, verify an nth-term formula, or compare how different sequences grow. That is useful in coursework, coding problems, and quick exploratory math, including checks against common patterns you might look up in OEIS or derive from difference tables. It also helps when you want the first few terms and the closed form side by side.
Arithmetic: aₙ = a₁ + (n-1)d, Sum = n/2 × (a₁ + aₙ). Geometric: aₙ = a₁ × r^(n-1), Sum = a₁(1-rⁿ)/(1-r). Fibonacci: Fₙ = Fₙ₋₁ + Fₙ₋₂. Triangular: Tₙ = n(n+1)/2. Square: Sₙ = n².
Result: Terms: 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536. Sum = 3,069
Geometric sequence with a₁ = 3, r = 2: each term is double the previous. The 10th term = 3 × 2⁹ = 1,536. Sum = 3 × (2¹⁰ - 1) / (2 - 1) = 3 × 1,023 = 3,069.
Arithmetic sequences model any uniformly increasing or decreasing pattern: equal monthly payments, clock ticks, temperature scales, and linear depreciation. Their simplicity makes them the first sequences students learn, but they appear frequently in real-world modeling.
Geometric sequences model exponential growth and decay: compound interest, population growth, radioactive decay, and signal attenuation. The ability of geometric sums to converge (when |r| < 1) makes them fundamental to finance (present value calculations), digital signal processing (z-transforms), and probability theory.
The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...) is arguably the most famous sequence in mathematics. The ratio of consecutive terms converges to φ = (1+√5)/2 ≈ 1.618, the golden ratio. This appears in art (golden rectangle), architecture (Parthenon proportions), nature (spiral shells, flower petals, pinecone scales), and algorithm analysis.
Fibonacci-type sequences (where aₙ = aₙ₋₁ + aₙ₋₂ with any starting values) all converge to the same ratio φ, regardless of starting values. This universality is why the golden ratio appears so ubiquitously.
For large n, direct computation of sequence terms can overflow standard number types. Techniques include: modular arithmetic (compute aₙ mod m without computing aₙ), matrix exponentiation (compute Fibonacci numbers in O(log n) time), and closed-form expressions (Binet's formula for Fibonacci, though it loses precision for large n due to floating-point issues).
A sequence where each term differs from the previous by a constant amount d (common difference). Example: 2, 5, 8, 11, 14... (d=3). The nth term is aₙ = a₁ + (n-1)d. The sum of n terms is n/2 × (first + last).
A sequence where each term is multiplied by a constant ratio r. Example: 3, 6, 12, 24... (r=2). The nth term is aₙ = a₁ × r^(n-1). If |r| < 1, the infinite sum converges to a₁/(1-r).
Each term is the sum of the two preceding: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... The ratio of consecutive terms approaches the golden ratio φ ≈ 1.618. It appears in nature (phyllotaxis), computer science (Fibonacci heaps), and financial analysis.
The number of dots in an equilateral triangle pattern: 1, 3, 6, 10, 15, 21... The nth triangular number is Tₙ = n(n+1)/2. Triangular numbers appear in combinatorics (n choose 2 = T(n-1)), bowling pin arrangements, and handshake problems.
If |r| < 1, yes: S∞ = a₁/(1-r). For example, 1 + 1/2 + 1/4 + 1/8 + ... = 2. If |r| ≥ 1, the series diverges (grows without bound). This convergence is fundamental to calculus, signal processing, and financial math (present value of perpetuities).
Check differences: if constant → arithmetic. Check ratios: if constant → geometric. Check second differences: if constant → quadratic. If aₙ = aₙ₋₁ + aₙ₋₂ → Fibonacci-type. For complex patterns, try OEIS.org to look up the integer sequence.