2026-08-07 10 min

Number Sequence Calculator

Arithmetic, geometric, and Fibonacci sequence calculator: nth term, full sequence, and partial sums with step-by-step formulas.

Calculator

Arithmetic Sequence

First number (a₁)

Common difference (f)

nth number (n)

Geometric Sequence

First number (a)

Common ratio (r)

nth number (n)

Fibonacci Sequence

nth number (n)

Results update automatically

Results update live as you type

Status:Arith aₙ = 19Geo aₙ = 512Fib = 55

Sequence Summary

Arithmetic aₙ = a₁ + f·(n−1)19

Seq: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19

Arithmetic sum100

Geometric aₙ = a·rⁿ⁻¹512

Seq: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512

Geometric sum1,023

Fibonacci F(10)55

Seq: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55

Results

Normal

Arithmetic nth Term

a₁ + f·(n − 1)

Arithmetic Sum

100

n·(a₁ + aₙ)/2

Geometric nth Term

512

a·rⁿ⁻¹

Geometric Sum

1,023

a·(1 − rⁿ)/(1 − r), or a·n if r = 1

Fibonacci F(n)

F₀=0, F₁=1, Fₙ=Fₙ₋₁+Fₙ₋₂

Fibonacci Terms Shown

F₀ through F(n) inclusive

Terms compute in double precision with n capped at 50 for readable, exact-enough lists. Geometric r = 1 uses the flat Sₙ = a·n rule.

Visualizations

Arithmetic Terms (first 8)

1a1

3a2

5a3

7a4

9a5

11a6

13a7

15a8

Arithmetic Progression Path

Terms Requested

Arithmetic n10

nth Term: Arithmetic vs Geometric

Arith19

Formula

Arithmetic sequences grow by repeated addition, geometric by repeated multiplication, and Fibonacci by self-reference. Each family has a direct nth-term rule and a summation identity, so any term and any partial sum follow without enumerating by hand.

aₙ = a₁ + f·(n−1) · Sₙ = n·(a₁+aₙ)/2 · gₙ = a·rⁿ⁻¹ · Fₙ = Fₙ₋₁+Fₙ₋₂

Alternative Form

Geometric sum: Sₙ = a·(1−rⁿ)/(1−r) (r≠1); infinite |r|<1 → a/(1−r)

Variables

Symbol	Name	Unit	Description
a₁, f	Arithmetic seed & step	number	First term and common difference
a, r	Geometric seed & ratio	number	First term and common ratio
n	Index	count	1-based for arith/geo (0-based option for Fibonacci)
Sₙ	Partial sum	number	Sum of the first n terms
Fₙ	Fibonacci term	number	F₀=0, F₁=1 recurrence

Explanation

Addition compounds linearly, multiplication exponentially, and Fibonacci recursively — three archetypes covering most elementary growth.

Derivation

Arithmetic: aₙ = a₁ + f·(n−1); Sₙ = n·(a₁+aₙ)/2 by endpoint pairing. Geometric: aₙ = a·rⁿ⁻¹; Sₙ = a·(1−rⁿ)/(1−r) by subtracting r·Sₙ from Sₙ. Fibonacci: Fₙ = Fₙ₋₁ + Fₙ₋₂ from the recurrence definition.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Arithmetic: step from the first term by the common difference.

aₙ = a₁ + f·(n − 1)

a1:1

f:2

n:10

Intermediate: 1 + 2·(10 − 1)

Result: aₙ = 19

Step 2

Arithmetic sum: average the endpoints and multiply by the count.

Sₙ = n·(a₁ + aₙ)/2

n:10

Intermediate: 10·(1 + 19)/2

Result: Sₙ = 100

Step 3

Geometric: scale the first term by the ratio raised to n − 1.

aₙ = a·rⁿ⁻¹

a:1

r:2

n:10

Intermediate: 1·2^(10 − 1)

Result: aₙ = 512

Step 4

Geometric sum: closed form, with the r = 1 flat case handled separately.

Sₙ = a·(1 − rⁿ)/(1 − r); r=1 → a·n

n:10

Intermediate: n=10

Result: Sₙ = 1,023

Step 5

Fibonacci: seed 0, 1 and accumulate pairwise sums up to F(n).

F₀=0, F₁=1, Fₙ=Fₙ₋₁+Fₙ₋₂

n:10

Intermediate: F₀…F(10)

Result: F(10) = 55

Step 6

Compare growth: linear (arithmetic) vs exponential (geometric) vs recursive (Fibonacci).

n: 10 / 10 / 10

Arith nth:19

Geo nth:512

Fib nth:55

Intermediate: Same index, three growth laws

Result: nth terms: 19 · 512 · 55

Final Answer

Worked Examples

See how the calculator works with different scenarios.

Scenario

Odd numbers starting at 1 with difference 2, first 10 terms.

Inputs

a1:1

f:2

n:10

Calculation

a₁₀ = 1 + 2·9 = 19 S₁₀ = 10·(1 + 19)/2 = 100 Seq: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19

Result

10th term 19 · sum 100

Interpretation

The classic odd-number triangle: summing the first 10 odds yields the perfect square 100.

Understanding Your Result

Your Result: Arithmetic nth Term: 19

Typical range: Arithmetic differences are usually small constants; ratios between 0 and 2 cover decay through doubling; Fibonacci n ≤ 50 stays exact and readable.

What This Means

Each panel answers three questions: which value sits at position n, what the terms look like in order, and what they add up to.

Why It Matters

Deposits, depreciation, populations, and algorithms all follow one of these three growth laws — naming the law predicts the future.

Professional Interpretation

Compare the nth terms at equal n: linear gaps stay constant, exponential gaps multiply, and Fibonacci gaps follow φ — the chart makes the regime obvious.

Recommendations

•Keep n ≤ 50 for readable lists
•Check r = 1 separately (flat geometric)
•Use sums for totals, terms for positions
•Watch exponential overflow past rⁿ ≈ 10¹⁵
•Quote Fibonacci with its index (F₁₀ = 55)

Educational Content

Introduction

Sequences are functions on the natural numbers. Arithmetic adds, geometric multiplies, and Fibonacci remembers — together they generate lines, exponentials, and self-similar growth.

Concept Explanation

Closed forms (aₙ directly from n) beat recursion for speed: arithmetic and geometric have them; Fibonacci has Binet’s form. Summation identities collapse n additions into one line.

Applications

▸Straight-line depreciation and level annuities

▸Compound interest, inflation, half-life decay

▸Algorithm analysis and recurrence relations

▸Population and epidemic modeling

▸Music intervals and tuning ratios

▸Tiling, spirals, and phyllotaxis

Benefits

•One rule predicts any term without listing
•Sums give totals in closed form
•Charts reveal linear vs exponential regimes
•Fibonacci links integers to the golden ratio
•OEIS connects patterns to literature

Limitations

•n capped at 50 for display and precision
•Formulas assume exact constant f/r
•Noisy real data needs fitting before modeling
•Huge geometric terms lose integer exactness past 2⁵³

Best Practices

•Identify difference vs ratio first
•State the index convention (F₀ vs F₁)
•Pair sums: endpoints for arithmetic, r·Sₙ trick for geometric
•Plot before extrapolating exponentials
•Verify missing terms by re-substituting

Common Misconceptions

Myth: All fast-growing sequences are geometric.

Truth: Fibonacci grows exponentially too (≈φⁿ) but by addition, not a fixed ratio — structure differs even when speed looks similar.

Myth: The nth term needs all previous terms.

Truth: Closed forms jump straight to n: arithmetic and geometric never need the middle; only naive Fibonacci does.

Myth: Series and sequence are synonyms.

Truth: Sequence lists, series sums. Convergence questions belong to series.

Myth: r = 1 breaks the geometric sum.

Truth: It just needs the separate rule Sₙ = a·n — a flat line, not an error.

Practical Tips

•Name f vs r before typing numbers
•Keep n within 1-50 (0-50 Fibonacci)
•Read the sum when asked for totals
•Compare nth terms at equal n
•Use the chart to spot linear vs exponential
•Quote Fibonacci as F(n) = value
•Re-run after any seed or step change

Common Mistakes

Off-by-one indexing (a₀ vs a₁)

This tool is 1-based for arith/geo and F₀-based for Fibonacci — match n to the stated convention.

Using the sum when asked for the term

aₙ is the position value; Sₙ is the running total. Read the label before copying.

Forgetting r = 1 is flat

Ratio exactly 1 repeats the seed: Sₙ = a·n, not the divided formula.

Extrapolating exponentials from two points

Two points fit many curves — confirm the ratio is constant across several steps first.

Pasting 50-term lists into reports

Quote nth + sum and link the page; reserve full listings for appendices.

Assumptions

This calculator makes the following assumptions to provide estimates:

Constant Step: f and r are fixed across all terms in each panel.

Index Conventions: Arith/geo are 1-based; Fibonacci uses F₀ = 0, F₁ = 1.

Display Cap: Sequences render at most 51 Fibonacci or 50 arith/geo terms.

Double Arithmetic: Terms compute in IEEE 754 doubles; integers exact to 2⁵³.

Local Only: Nothing is uploaded; refresh resets the panels.

Limitations

Be aware of these limitations when using this calculator:

Three Families: Squares, cubes, harmonic, and custom recurrences are out of scope.

No Convergence Engine: Infinite-series limits beyond |r| < 1 geometric are not evaluated.

Precision Ceiling: Huge terms (rⁿ, F₅₀) round past integer exactness.

Illustrative Chart: The SVG scales to the arithmetic window; steep geometric runs clamp.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a number sequence?

An ordered list of numbers following a rule. Each entry is a term. Sequences can be finite (stopping) or infinite (continuing), and they model growth, patterns, and relationships across math and science.

Arithmetic vs geometric vs Fibonacci?

Arithmetic adds a constant difference each step (linear growth). Geometric multiplies by a constant ratio (exponential growth/decay). Fibonacci adds the two previous terms starting from 0, 1 (recursive growth toward the golden ratio).

What is the arithmetic nth-term formula?

aₙ = a₁ + f·(n − 1), where a₁ is the first term and f the common difference. Example: 5, 8, 11, … has a₁ = 5, f = 3, so a₁₀ = 5 + 3·9 = 32.

What is the arithmetic sum formula?

Sₙ = n·(a₁ + aₙ)/2. Pairing first+last terms always gives the same total — Gauss famously used this to sum 1…100 as 100·101/2 = 5050.

What is the geometric nth-term formula?

aₙ = a·rⁿ⁻¹ with first term a and ratio r. Example: 2, 6, 18, … has a = 2, r = 3, so a₆ = 2·3⁵ = 486.

What is the geometric sum formula?

Sₙ = a·(1 − rⁿ)/(1 − r) for r ≠ 1; Sₙ = a·n when r = 1. Infinite series with |r| < 1 converge to a/(1 − r), e.g. 1 + ½ + ¼ + … = 2.

How does Fibonacci work?

F₀ = 0, F₁ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂. Terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … Ratios Fₙ/Fₙ₋₁ approach φ ≈ 1.618.

What is the golden ratio?

φ = (1 + √5)/2 ≈ 1.61803. Consecutive Fibonacci ratios converge to it. It appears in art, architecture, phyllotaxis, and Binet’s closed form.

What is Binet's formula?

Fₙ = (φⁿ − ψⁿ)/√5 with φ = (1+√5)/2 and ψ = (1−√5)/2. It computes Fibonacci directly without recursion.

Sequence vs series?

A sequence lists terms (2, 4, 6, 8); a series sums them (2+4+6+8 = 20). Infinite series may converge to a finite limit or diverge.

What is the common difference?

The constant added each step in an arithmetic sequence. Subtract any term from its successor: in 7, 5, 3, 1, … the difference is −2. Zero gives a constant sequence.

What is the common ratio?

The constant multiplier in a geometric sequence. Divide any term by its predecessor: 100, 50, 25, … has r = 0.5. |r| < 1 decays; r > 1 grows.

Why cap n at 50 terms?

Geometric and Fibonacci terms explode (2⁵⁰ ≈ 10¹⁵, F₅₀ ≈ 10¹⁰) and long lists hurt readability. Fifty terms show the pattern without overflow or layout breakage.

What are triangular numbers?

Tₙ = n(n+1)/2: 1, 3, 6, 10, 15, … They count triangular arrangements and equal the sum 1…n.

What are square and cube numbers?

Squares n²: 1, 4, 9, 16, … Cubes n³: 1, 8, 27, 64, … Consecutive squares differ by odd numbers; cubes sum to the squared triangular total.

What is sigma notation?

Σₖ₌₁ⁿ aₖ means a₁ + … + aₙ. Example: Σₖ₌₁⁵ k² = 55. It is linear: Σ(caₖ + bₖ) = cΣaₖ + Σbₖ.

How do I find a missing term?

Check first differences (arithmetic?), ratios (geometric?), then second differences or known shapes (squares, Fibonacci). Example: 2, 5, ?, 11, 14 has difference 3, so 8.

What is a recursive definition?

Initial terms plus a rule from predecessors: arithmetic a₁ = 3, aₙ = aₙ₋₁ + 4; geometric g₁ = 2, gₙ = 3·gₙ₋₁; Fibonacci F₀ = 0, F₁ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂.

What is the OEIS?

The On-Line Encyclopedia of Integer Sequences (350,000+ entries). Paste unknown terms there to identify the pattern, formulas, and references.

Where do sequences appear in nature?

Fibonacci counts appear in sunflower spirals, petal numbers (3, 5, 8, 13, 21, 34), pinecones, and shells — efficient packing produced by growth rules.

How are sequences used in finance?

Arithmetic models straight-line depreciation and level payments; geometric models compound interest and inflation. Annuity present value is a geometric sum.

How are sequences used in computing?

Arrays are sequences; recursion and divide-and-conquer follow sequence patterns; Fibonacci heaps, sequence alignment (DNA), and time-series forecasting all build on them.

What is a convergent sequence?

Terms approaching a finite limit, e.g. 1, ½, ⅓, … → 0. Bounded monotone sequences always converge; others may diverge or oscillate.

Does this tool store my sequences?

No. All computation is local JavaScript; inputs and results never leave the browser.