Number Sequence Calculator

This number sequence calculator works as an arithmetic sequence calculator, geometric sequence calculator, Fibonacci calculator, and nth term calculator in one tool. Paste any list of numbers and it finds the pattern, predicts the next five terms, and returns the sum when the rule is arithmetic. It handles the three most common sequence types taught in algebra and covered in most series calculator searches.

1. Pick the sequence type (auto-detected). The tool checks arithmetic progression first, then geometric, then Fibonacci-like. You do not need to label it yourself.
2. Enter the first term and the rest. Type comma- separated numbers. Three terms is the minimum, five or more gives a confident detection and sharper next-term predictions.
3. Read the common difference or common ratio. For an arithmetic sequence the tool returns d (the constant you add). For a geometric sequence it returns r (the constant you multiply by).
4. Choose how many terms to project. The next five terms are calculated automatically using the nth term formula shown in the result card.
5. Check the sum. For arithmetic sequences the calculator adds every entered term using the closed form Sₙ = n/2 × (a₁ + aₙ), which is the same output a sum of sequence calculator returns.

Preset buttons load an arithmetic example (3, 7, 11, 15, 19), a geometric example (2, 4, 8, 16, 32), the Fibonacci sequence, and the square numbers so you can see how each rule behaves before entering your own data.

Every sequence type has its own nth term formula and its own way of totalling the first n terms. Below are the formulas the calculator uses internally, each with a worked example you can verify by hand.

Arithmetic Sequence

An arithmetic progression adds the same common difference d between consecutive terms. The arithmetic sequence calculator uses this closed form for the nth term:

aₙ = a₁ + (n − 1) × d

Sum of first n terms:
Sₙ = n/2 × (a₁ + aₙ) = n/2 × (2a₁ + (n − 1)d)

Worked example: the sequence 2, 5, 8, 11, …

a₁ = 2, d = 3
10th term: a₁₀ = 2 + 9 × 3 = 29
Sum of first 10 terms: S₁₀ = 10/2 × (2 + 29) = 5 × 31 = 155

Geometric Sequence

A geometric sequence multiplies by the same common ratio r each step. The geometric sequence calculator applies the standard formula:

aₙ = a₁ × r^(n − 1)

Sum of first n terms (for r ≠ 1):
Sₙ = a₁ × (1 − rⁿ) / (1 − r)

Worked example: the sequence 3, 6, 12, 24, …

a₁ = 3, r = 2
8th term: a₈ = 3 × 2⁷ = 3 × 128 = 384
Sum of first 8 terms: S₈ = 3 × (1 − 256) / (1 − 2) = 3 × 255 = 765

Infinite Geometric Sum

When |r| < 1 the terms shrink fast enough that the total converges to a finite number. This is the formula every sum of sequence calculator uses for infinite geometric series:

S∞ = a₁ / (1 − r)    (valid only when |r| < 1)

Worked example: 1 + 1/2 + 1/4 + 1/8 + 1/16 + …
a₁ = 1, r = 1/2
S∞ = 1 / (1 − 0.5) = 2

Zeno's paradox of the runner covering half the remaining distance forever is the same sum. The total distance is finite even though the number of steps is not.

Fibonacci Sequence

The Fibonacci calculator uses a recursive rule rather than a simple closed form. Each term equals the sum of the two before it:

F₁ = 1, F₂ = 1
Fₙ = Fₙ₋₁ + Fₙ₋₂  for n ≥ 3

First 12 terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

The ratio of consecutive Fibonacci numbers converges to the golden ratio φ = (1 + √5)/2 ≈ 1.618. For example 144/89 ≈ 1.6180, and every further ratio matches φ more tightly:

Fₙ₊₁ / Fₙ → φ = (1 + √5) / 2 ≈ 1.6180339887

Binet's closed form:
Fₙ = (φⁿ − ψⁿ) / √5,    where ψ = (1 − √5) / 2 ≈ −0.618

Other Named Sequences

• Triangular numbers: Tₙ = n(n + 1) / 2 → 1, 3, 6, 10, 15, 21, …
• Square numbers: Sₙ = n² → 1, 4, 9, 16, 25, 36, …
• Cube numbers: Cₙ = n³ → 1, 8, 27, 64, 125, 216, …
• Factorials: n! = 1 × 2 × 3 × … × n → 1, 2, 6, 24, 120, 720, …
• Powers of 2: 2ⁿ⁻¹ → 1, 2, 4, 8, 16, 32, 64, 128, …

Quick-Reference Table: First 10 Terms of Common Sequences

Notice how the geometric column blows past the others by term 8. That is the signature of exponential growth: doubling beats any arithmetic progression once n is large enough.

Sequences are not just an algebra exercise. Arithmetic and geometric progressions describe money, biology, music, and screen hardware. Knowing which kind of sequence you are looking at tells you whether a quantity will grow linearly, explode exponentially, or level off.

Where Sequences Appear in Everyday Math

• Compound interest is a geometric sequence. A $1,000 balance at 6% annual interest becomes 1,000 × 1.06ⁿ after n years: $1,060 after 1 year, $1,338.23 after 5, $1,790.85 after 10. The common ratio is r = 1.06.
• Loan amortization mixes both. The monthly payment is fixed (arithmetic pattern of payment dates), but the split between interest and principal follows a geometric decay because interest is always a fixed percent of the shrinking balance.
• Population growth. Bacteria dividing every 20 minutes follow a geometric sequence with r = 2. Starting from 1,000 cells you have 1,024,000 after just over three hours.
• Musical frequencies are geometric. Each octave doubles the frequency, so the A notes on a piano form the sequence 55, 110, 220, 440, 880, 1760 Hz with r = 2. The 12 semitones in between divide that doubling evenly, giving r = 2^(1/12) ≈ 1.0595 between adjacent keys.
• Pixel rows on a tapered screen. Some curved or trapezoid displays use arithmetic progressions for pixel counts per row (row 1 has 1,080 pixels, each subsequent row drops by a fixed d). Sum formulas tell engineers the total pixel count instantly.
• Stadium seating. Rows of seats where each row adds 2 more than the last is a textbook arithmetic example. Twenty rows starting at 30 seats and growing by 2 per row hold 20/2 × (30 + 68) = 980 seats.

Fibonacci and the Golden Ratio in Nature

Fibonacci numbers do show up in nature, though the topic is frequently overstated. The real cases are concentrated in plant biology where the golden angle (≈137.5°, derived from φ) maximises exposure to sunlight and packing efficiency for new growth.

• Sunflower seed heads arrange seeds in two sets of interlocking spirals. The counts are almost always consecutive Fibonacci numbers: 34 and 55, or 55 and 89.
• Pinecone bracts and pineapple scales follow the same 8 and 13, or 13 and 21 spiral counts.
• Nautilus shells are often cited as Fibonacci spirals, but they are technically logarithmic spirals with a growth factor close to but not equal to φ. The resemblance is real, the exact Fibonacci match is a myth.
• Branching trees and leaf arrangements (phyllotaxis) on species like elm, oak, and pear use Fibonacci ratios to space leaves around a stem so higher leaves do not shade lower ones.

Claims about Fibonacci in the human face, Greek architecture, and stock charts are mostly pattern-matching on loose data. The plant biology cases are genuine and well documented.

How to Tell Which Kind of Sequence You Have

Given any list of numbers, work through this quick decision tree before reaching for a general series calculator:

OEIS: When Your Sequence Is Not Arithmetic, Geometric, or Fibonacci

The Online Encyclopedia of Integer Sequences (OEIS) is the single best reference for identifying an unknown sequence. It catalogues more than 370,000 sequences, each with its formula, known occurrences, and references. Type any five or six terms into the search box at oeis.org and it will tell you whether those numbers match a known pattern.

Example: enter "1, 2, 5, 14, 42, 132" and OEIS returns the Catalan numbers, which count things like binary tree shapes and non-crossing handshakes around a round table. No amount of arithmetic-ratio checking would surface that name. OEIS turns an unknown sequence into a one-search lookup.