Number Sequence Calculator

Generate and analyse arithmetic, geometric, Fibonacci, and custom number sequences. Find the n-th term, full sequence, cumulative sum, formulas, and detailed step-by-step calculations.

Arithmetic Sequence Calculator

definition: aₙ = a₁ + d · (n − 1)

example: 1, 3, 5, 7, 9, 11, 13 …

The first number (a₁)

Common difference (d)

The n-th number to obtain

Common Sequences Explorer

Generate the first n terms of a library sequence.

Terms

What Is A Number Sequence?

A number sequence is an ordered list of numbers governed by a rule. Every term has a fixed position — a₁, a₂, a₃, … aₙ — and the rule lets you predict the next term, the n-th term, and the cumulative sum without writing every value out. Some rules are explicit (each term is a direct function of n); others are recursive (each term depends on one or more previous terms).

This calculator handles the four most common patterns — arithmetic, geometric, Fibonacci, and a free-form custom analyser that detects quadratic, cubic, exponential, and Fibonacci-like progressions automatically — plus library presets for prime, triangular, square, cube, Lucas, and Catalan numbers. Each calculation returns the full sequence, the n-th value, the running sum, every formula used, a worked solution, and a chart. It pairs naturally with the mean median mode range calculator, the standard deviation calculator, and the scientific calculator.

How Sequence Calculations Work

Identify the rule

Inspect the first few terms. A constant difference between consecutive terms means arithmetic; a constant ratio means geometric; a term that depends on two predecessors means Fibonacci-like. Polynomial sequences show a constant second or third difference.

Apply the explicit formula

Once the rule is known, the n-th term can be computed in O(1). Arithmetic uses aₙ = a₁ + d(n − 1); geometric uses aₙ = a₁ · r^(n − 1); Fibonacci is naturally recursive but also has Binet's closed form Fₙ = (φⁿ − ψⁿ) / √5.

Sum with a closed form

Brute-force addition works but a closed form is faster and more accurate for large n. Arithmetic sum: Sₙ = n/2 · [2a₁ + (n − 1)d]. Geometric sum: Sₙ = a₁(rⁿ − 1) / (r − 1). Fibonacci partial sum: Sₙ = Fₙ₊₂ − 1.

Verify and interpret

Every calculator returns the full sequence, the n-th value, and the cumulative sum side-by-side with the formula used. Verify by spot-checking a small case by hand, then trust the engine for larger n.

6 Ways to Use This Number Sequence Calculator

Homework + practice

Solve any arithmetic, geometric, or Fibonacci problem step-by-step. The calculation tab mirrors the way the answer would be written out in a textbook, so the working can be copied straight onto paper.

Find the n-th term

Skip the manual recursion. Plug in a₁, d (or r), and n, and the calculator returns aₙ instantly using the closed-form formula — accurate even for n = 100.

Compute partial sums

The cumulative sum Sₙ uses the closed-form identity for every supported pattern, so you avoid float-error accumulation that brute-force addition would introduce on large sequences.

Detect unknown patterns

Paste any list of numbers into the Custom tab. The pattern detector checks first differences (arithmetic), ratios (geometric), second / third differences (quadratic / cubic), and Fibonacci-like recursion.

Explore famous sequences

The Common Sequences Explorer generates the first n primes, triangular, square, cube, Lucas, and Catalan numbers on demand — handy for combinatorics, number-theory, and discrete-maths coursework.

Project growth + decay

Geometric sequences model compound interest, population growth, radioactive decay, and bacterial replication. Plug in the multiplier, the starting value, and the number of periods to see the trajectory.

Best Practices For Working With Sequences

Always inspect the first 5 to 10 terms before picking a formula. A sequence that looks arithmetic can be quadratic in disguise — the difference between adjacent terms grows linearly rather than staying constant. Compute first differences, then second differences. Two rows of constant differences confirm a quadratic; three rows confirm a cubic. The Custom Sequence tab on this page automates that check.

Prefer the closed-form expression when n is large. Computing the 80-th term by stepping a Fibonacci recursion in plain JavaScript is fine, but the same recursion in spreadsheet form can drift due to floating-point error after a few dozen terms. This calculator uses BigInt under the hood for Fibonacci, Lucas, and Catalan numbers so every digit stays exact up through n = 100.

Sanity-check the sign of the common difference or ratio before you trust the trajectory. A negative d means an arithmetic sequence decreases without bound; |r| > 1 means a geometric sequence diverges; |r| < 1 means it converges to zero. The Statistics card reports monotonicity and convergence on every calculation so the behaviour is visible at a glance.

Why Number Sequences Matter

Finance & compound growth

Compound interest, dividend growth, recurring deposits, and amortisation schedules are geometric sequences in disguise. The future value of a fixed-rate deposit after n compounding periods is just a₁ · (1 + r)ⁿ — the n-th term of a geometric sequence.

Science & decay

Radioactive decay, drug elimination, and capacitor discharge follow geometric (decay) sequences. Population growth and bacterial doubling follow geometric (growth) sequences. Once the rate is known, sequence formulas project any future value.

Computer science

Algorithm complexity is described by sequences — bubble sort is O(n²), binary search is O(log₂ n), recursive Fibonacci without memoisation runs in O(φⁿ). Recurrence relations and dynamic programming are sequence problems at heart.

Mathematics curriculum

Sequences appear in every high-school and undergraduate maths course — discrete maths, calculus, combinatorics, number theory, and probability. Mastering the four basic patterns and the difference-detection trick unlocks most of them.

Where Sequence Problems Get Tricky

Polynomial in disguise

A constant second difference (not first) means the sequence is quadratic. A constant third difference means cubic. The Custom Sequence tab applies this finite-difference check automatically and returns the polynomial coefficients.

Geometric with r = 1

When r = 1, every term equals a₁ and the closed-form sum formula contains a divide-by-zero. This calculator falls back to Sₙ = a₁ · n in that limit case so the answer remains finite.

Infinite geometric series

When |r| < 1, the infinite partial sums converge to S∞ = a₁ ÷ (1 − r). When |r| ≥ 1 the series diverges. Read the convergence label on the Statistics card to see which regime your sequence sits in.

Indexing conventions

Most textbooks start arithmetic and geometric sequences at a₁. Fibonacci is often indexed from F₀ = 0; this calculator uses the F₁ = F₂ = 1 convention so the 10th value equals 55, matching the screenshot reference.

The Core Sequence Formulas

Arithmetic n-th term

aₙ = a₁ + d · (n − 1)

Straight-line growth in steps of d.

Arithmetic sum

Sₙ = n/2 · [2a₁ + (n − 1)d]

Gauss's pairing trick — average × count.

Geometric n-th term

aₙ = a₁ · r^(n − 1)

Multiplicative growth, exponent counts steps.

Geometric sum

Sₙ = a₁(rⁿ − 1) / (r − 1)

Closed form when r ≠ 1.

Infinite geometric sum

S∞ = a₁ / (1 − r), |r| < 1

Converges when the ratio's absolute value is below 1.

Fibonacci recurrence

Fₙ = Fₙ₋₁ + Fₙ₋₂, F₁ = F₂ = 1

Each term is the sum of the previous two.

Fibonacci Binet form

Fₙ = (φⁿ − ψⁿ) / √5

φ = (1 + √5)/2, ψ = (1 − √5)/2.

Fibonacci partial sum

Sₙ = Fₙ₊₂ − 1

Beautiful identity — saves a loop.

Triangular number

Tₙ = n(n + 1) / 2

Sum of the first n positive integers.

Square number

Sₙ = n²

Sum of the first n odd integers.

Cube number

Cₙ = n³

Σ Cₖ = (Tₙ)².

Catalan number

Cₙ = (2n)! / [(n + 1)! · n!]

Counts binary trees, parenthesisations, paths.

Common Sequence Mistakes

1
Off-by-one indexing
Most explicit formulas use (n − 1) in the exponent or multiplier — not n. The 1st term is a₁ + d · 0 in arithmetic and a₁ · r⁰ in geometric. Forgetting the −1 shifts every term.
2
Confusing arithmetic with geometric
Arithmetic sequences add the same number each step; geometric sequences multiply by the same factor. Inspect both first differences and first ratios before committing to a formula.
3
Brute-forcing the partial sum
Computing Sₙ by adding up every term works but invites float-error drift on large n. The closed-form sum formulas in this calculator avoid the issue completely.
4
Trusting recurrence at scale
Naïve Fibonacci recursion is exponential time. Use the iterative form (or this calculator) for any n above ~30 — the recursive call tree explodes otherwise.
5
Misreading the Fibonacci start
Different texts index Fibonacci from F₀ = 0 or F₁ = 1 — both are valid. The 10th value is 34 in the first convention and 55 in the second. Always check which convention a problem uses.
6
Stopping at first differences
A non-constant first-difference row does not mean the sequence is exotic. Take a second (or third) difference. A constant row at depth k indicates a polynomial of degree k.

Real-Life Uses Of Number Sequences

Compound interest + SIPs

Every recurring-deposit, SIP, or compound-interest projection rests on geometric sequences. The balance after k years equals P · (1 + r)ᵏ — the k-th term of the geometric sequence starting at P with ratio (1 + r).

Population + decay models

Exponential growth and decay are geometric sequences with discrete time steps. Microbial cultures double, isotopes halve, and the same formula projects both — just with different ratios.

Combinatorics + tilings

Catalan numbers count balanced parentheses, binary search trees, and triangulations of polygons. Triangular and square numbers count dots arrangeable into shapes. These appear constantly in discrete-maths problems.

Algorithm complexity

Big-O notation is a sequence comparison. Linear O(n), quadratic O(n²), cubic O(n³), and exponential O(2ⁿ) all describe how an algorithm's running time grows as input size n increases.

Methodology you can verify

Every n-th term and partial sum is computed from the canonical closed-form expressions. Fibonacci, Lucas, and Catalan numbers use BigInt arithmetic so every digit stays exact through n = 100. Pattern detection runs first-, second-, and third-difference checks plus ratio tests in the order arithmetic → geometric → quadratic → cubic → Fibonacci-like → exponential, returning the most specific match. Read more on the methodology and editorial policy pages.

Frequently Asked Questions

A number sequence is an ordered list of numbers governed by a fixed rule. Each value has a position — a₁, a₂, a₃, … aₙ — and the rule lets you compute the next term, the n-th term, and the cumulative sum without writing out every value. Common families include arithmetic sequences (each term is the previous plus a constant), geometric sequences (each term is the previous times a constant), Fibonacci-like sequences (each term is the sum of the previous two), and polynomial sequences like triangular and square numbers. This calculator handles all four and auto-detects the polynomial degree on free-form lists.

An arithmetic sequence is one in which every consecutive pair of terms differs by the same constant, called the common difference d. The general (explicit) formula is aₙ = a₁ + d · (n − 1), where a₁ is the first term. Examples: 1, 3, 5, 7, 9, … (a₁ = 1, d = 2) and 2, 7, 12, 17, 22, … (a₁ = 2, d = 5). For the second example, the 20th term is 2 + 5 · 19 = 97 and the partial sum is S₂₀ = 20/2 · [2(2) + 19(5)] = 10 · 99 = 990.

A geometric sequence is one in which every consecutive pair of terms shares the same ratio, called the common ratio r. The general formula is aₙ = a₁ · r^(n − 1). Examples: 1, 2, 4, 8, 16, … (a₁ = 1, r = 2) and 2, 10, 50, 250, … (a₁ = 2, r = 5). For the second example, the 12th term is 2 · 5¹¹ = 97 656 250 and the partial sum is 122 070 312, both returned by this calculator with the closed-form Sₙ = a₁(rⁿ − 1) / (r − 1).

The Fibonacci sequence is defined by F₁ = 1, F₂ = 1, and Fₙ = Fₙ₋₁ + Fₙ₋₂ for every n ≥ 3 — each term is the sum of the previous two. The first few values are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … so the 10th value is 55. Fibonacci numbers appear in nature (sunflower seed spirals, pinecone scales, nautilus shells), in mathematics (golden-ratio convergence — consecutive ratios approach φ ≈ 1.618), in art and architecture, and in computer science (Fibonacci heaps, dynamic programming examples).

Use the explicit formula for the pattern. Arithmetic: aₙ = a₁ + d · (n − 1). Geometric: aₙ = a₁ · r^(n − 1). Fibonacci: build it up from F₁ = F₂ = 1 — or use Binet's closed form Fₙ = (φⁿ − ψⁿ) / √5 where φ = (1 + √5)/2 and ψ = (1 − √5)/2. Triangular: Tₙ = n(n + 1) / 2. Square: Sₙ = n². Cube: Cₙ = n³. This calculator applies the right formula automatically once you pick the tab.

Use the closed-form sum. Arithmetic: Sₙ = n/2 · [2a₁ + (n − 1)d] (Gauss's pairing trick). Geometric: Sₙ = a₁(rⁿ − 1) / (r − 1) when r ≠ 1, or Sₙ = a₁ · n when r = 1. Fibonacci: Sₙ = Fₙ₊₂ − 1 — a beautiful identity that skips the loop entirely. Triangular sum: n(n + 1)(n + 2) / 6. Sum of squares: n(n + 1)(2n + 1) / 6. Sum of cubes: [n(n + 1) / 2]² (the square of the n-th triangular number). The calculator shows every substitution so you can verify the working.

The common difference d is the constant value added to each term of an arithmetic sequence to get the next term. It can be positive (sequence grows), negative (sequence shrinks), or zero (every term equals a₁). To find d from a sequence, subtract any term from the next: d = aₙ₊₁ − aₙ. If that subtraction is not constant across the whole sequence, the sequence is not arithmetic — try the Custom tab on this page to detect the actual pattern.

The common ratio r is the constant factor multiplied into each term of a geometric sequence to get the next term. r can be any non-zero real number — positive r preserves signs, negative r alternates signs, |r| > 1 makes the sequence diverge, |r| < 1 makes it converge to zero, and r = 1 produces a constant sequence. To find r from a sequence, divide any term by the previous: r = aₙ₊₁ / aₙ. If that ratio is not constant, the sequence is not geometric.

Sequences model almost every quantitative growth or decay process. Compound interest and SIPs follow geometric sequences (balance = P · (1 + r)ⁿ). Population growth and bacterial doubling follow exponential / geometric sequences. Radioactive decay and drug elimination follow geometric (decay) sequences. Linear depreciation, weekly savings goals, and constant-payment schedules follow arithmetic sequences. Computer-science algorithm complexity — O(n), O(n²), O(2ⁿ) — is described by sequence orders. Combinatorics counts use Catalan, Fibonacci, and binomial sequences.

An explicit formula (also called a closed form) computes aₙ directly from n in one step — for example aₙ = a₁ + d · (n − 1) for arithmetic. A recursive formula defines aₙ in terms of previous values — for example Fₙ = Fₙ₋₁ + Fₙ₋₂ for Fibonacci. Explicit formulas are faster and stable for large n; recursive formulas are often shorter to write but require building the sequence up term-by-term. Most sequences have both — Binet's formula is the explicit form of Fibonacci, even though the recursion is usually taught first.

It checks several pattern types in order. First, the first differences: if they are all equal, the sequence is arithmetic. Second, the first ratios: if they are all equal, it is geometric. Third, the second differences: if constant, the sequence is quadratic — and the calculator extracts the polynomial coefficients automatically. Fourth, the third differences: if constant, the sequence is cubic. Fifth, a Fibonacci-like check: if each term equals the sum of the previous two, it is a Lucas-style recurrence. Last, an exponential check on the log-spacing. The first pattern that matches wins.

The golden ratio φ = (1 + √5) / 2 ≈ 1.61803 is the limit of the ratio of consecutive Fibonacci numbers: lim Fₙ₊₁ / Fₙ = φ. It also appears in Binet's closed form Fₙ = (φⁿ − ψⁿ) / √5 (where ψ = 1 − φ ≈ −0.618). The golden ratio shows up in geometry (regular pentagons), art (the Parthenon, da Vinci's Vitruvian Man), and nature (sunflower spirals, nautilus shells) — all places where the Fibonacci sequence appears for the same reason.

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