Number Sequence Calculator: nth Term

Step	Detail
Formula	aₙ = a₁ + (n−1)d
Substitute	aₙ = 2 + (10−1)×3
Simplify	aₙ = 2 + 27 = 29
Sum Sₙ	Sₙ = n/2 × (2a₁ + (n−1)d) = 10/2 × (4 + 27) = 155

Step

Detail

Formula

aₙ = a₁ + (n−1)d

Substitute

aₙ = 2 + (10−1)×3

Simplify

aₙ = 2 + 27 = 29

Sum Sₙ

Sₙ = n/2 × (2a₁ + (n−1)d) = 10/2 × (4 + 27) = 155

Using the Number Sequence Calculator

Select Arithmetic, Geometric, or Fibonacci, enter the parameters (first term, common difference or ratio, or term index n), and use the Result pane for the n-th term and sum. Trusted by students and educators for precalculus and discrete math. The Logic Trace shows the formula steps; the sequence bar shows the first 10 terms. All calculations run locally.

Quick tips

Arithmetic

a_n = a_1 + (n-1)d

Enter first term (a₁) and common difference (d). Use target n for the term index

Geometric

•

a_n = a_1 \cdot r^{n-1}

Enter first term (a₁) and common ratio (r). If

•r

•< 1, the calculator shows the infinite sum S∞

Fibonacci

F₁ = 1, F₂ = 1.

F_n = F_{n-1} + F_{n-2}

Enter only the term index n. Large n gives very large terms (BigInt)

Copy

"Copy first 20 terms" copies the first 20 terms to the clipboard, separated by commas.

Number Sequence Calculator: Arithmetic, Geometric & Fibonacci

Find the n-th term and sum of arithmetic, geometric, and Fibonacci sequences. Arithmetic sequence formula with steps, geometric sequence ratio calculator, sum of arithmetic series. Logic Trace and visual sequence bar. Trusted by students and educators. No sign-up—all calculations run locally.

What This Calculator Does

Purpose

Key formulas

$a_n = a_1 + (n-1)d$

$a_n = a_1 \cdot r^{n-1}$

$F_n = F_{n-1} + F_{n-2}$
Features

Tabbed input for Arithmetic, Geometric, and Fibonacci. Inputs: first term (a₁), common difference (d) or common ratio (r), and target term index n. Result pane shows n-th term and Sₙ; Formula box (or last additions for Fibonacci); Copy first 20 terms to clipboard. All calculations run in your browser; no data is sent to servers.

This number sequence calculator computes the n-th term (aₙ or Fₙ) and the sum of the first n terms (Sₙ) for three sequence types: Arithmetic (constant difference d), Geometric (constant ratio r), and Fibonacci (each term is the sum of the two before it). It shows a step-by-step Logic Trace and a visual bar of the first 10 terms, with +d or ×r labels for arithmetic and geometric sequences. For geometric sequences with |r| < 1, it also shows the sum to infinity S∞. Students and teachers use it to check homework and to see the arithmetic sequence formula with steps or to find the n-th term of Fibonacci; the Logic Trace shows every substitution so you can verify results.

The Difference Between Arithmetic and Geometric Sequences

Arithmetic sequence: Each term is the previous term plus a fixed number d (the common difference).

a_n = a_1 + (n-1)d

Example: 2, 5, 8, 11,… (a₁ = 2, d = 3). The sum of an arithmetic series (first n terms):

S_n = \frac{n}{2}(2a_1 + (n-1)d)

Geometric sequence: Each term is the previous term multiplied by a fixed number r (the common ratio).

a_n = a_1 \cdot r^{n-1}

Example: 2, 6, 18, 54,… (a₁ = 2, r = 3). The sum of a geometric series (first n terms):

S_n = \frac{a_1(1-r^n)}{1-r}

when r ≠ 1. If |r| < 1, the infinite geometric series converges:

S_\infty = \frac{a_1}{1-r}

This calculator acts as a geometric sequence ratio calculator: enter a₁ and r to find any term and the sum (or S∞ when |r| < 1)

Fibonacci Sequence and the n-th Term

The Fibonacci sequence starts with F₁ = 1, F₂ = 1, and each later term is the sum of the two preceding:

F_n = F_{n-1} + F_{n-2}

So 1, 1, 2, 3, 5, 8, 13, 21,… To find the n-th term of Fibonacci, you can use this calculator: enter the term index n (1 to 500) and read Fₙ and the sum of the first n Fibonacci numbers (which equals Fₙ₊₂ − 1). Fibonacci numbers grow quickly; the calculator uses exact integer arithmetic so there are no rounding errors

How to Find the Common Difference in a Sequence

If you have a list of terms and suspect they form an arithmetic sequence, the common difference d is the constant gap:

d = a_2 - a_1

(or aₙ₊₁ − aₙ for any consecutive pair). Once you know a₁ and d, the n-th term is

a_n = a_1 + (n-1)d

This calculator skips the “find d from a list” step: you enter the first term and common difference directly, then choose n. The Logic Trace shows the arithmetic sequence formula with steps—substitution and simplification—so you can follow the algebra and use the result for homework or to check your own work

Number Sequence Calculator FAQ

? What is the n-th term of a sequence?

The n-th term (written aₙ or Fₙ for Fibonacci) is the value of the sequence at position n. In an arithmetic sequence:

a_n = a_1 + (n-1)d

In a geometric sequence:

a_n = a_1 \cdot r^{n-1}

In the Fibonacci sequence, F₁ = 1, F₂ = 1, and

F_n = F_{n-1} + F_{n-2}

This number sequence calculator finds the n-th term and the sum of the first n terms for all three types

? Can the common ratio in a geometric sequence be negative?

Yes. The common ratio (r) in a geometric sequence can be negative. When r is negative, terms alternate in sign (e.g. 2, −6, 18, −54…). The formulas

a_n = a_1 \cdot r^{n-1}

and the sum

S_n = \frac{a_1(1-r^n)}{1-r}

still apply. If |r| < 1, the infinite sum

S_\infty = \frac{a_1}{1-r}

converges; this calculator shows S∞ when |r| < 1

? Why is the Fibonacci sequence found in nature?

The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13…) appears in many natural patterns—petal counts, pinecone spirals, sunflower seeds, and branching—because growth that adds new “units” based on the previous two steps often produces Fibonacci numbers. The ratio of consecutive terms approaches the golden ratio φ ≈ 1.618. This calculator lets you find any Fₙ and the sum of the first n Fibonacci numbers.

? How do I find the common difference in an arithmetic sequence?

The common difference (d) is the fixed amount added each time:

d = a_2 - a_1

(or aₙ₊₁ − aₙ for any consecutive terms). Once you have d and the first term a₁, the n-th term is

a_n = a_1 + (n-1)d

In this calculator, enter a₁ and d and the target n; the Logic Trace shows the formula with your numbers step by step

? What is the sum of an arithmetic series?

The sum of the first n terms of an arithmetic sequence is

S_n = \frac{n}{2}(2a_1 + (n-1)d)

or equivalently Sₙ = n/2 × (first term + last term). This number sequence calculator displays Sₙ for arithmetic, geometric (including infinite series when |r| < 1), and Fibonacci sequences

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