Arithmetic Sequences

Sequence Setup

a₁₀ = 3 + (10−1)·4

First Term (a₁)

Common Diff (d)

Term Number (n)

Computes aₙ = a₁ + (n−1)d and lists the first 10 terms. n must be ≥ 1.

Examples

d = (a₇ − a₃) / (7 − 3)

Value of aₘ

Position m

Value of aₙ

Position n

Given two terms and their positions, finds d = (aₙ − aₘ)/(n − m), then back-solves for a₁.

Examples

d = (a₈ − a₁) / (8 − 1)

First Term (a₁)

Last Term (aₙ)

Number of Terms (n)

Given the first term, last term, and how many terms, find d and list every term. n must be ≥ 2.

Examples

Result

Enter values above and press Calculate to find the term, sequence, and explicit formula.

Answer

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Explicit Formula

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Sequence Terms

Arithmetic Sequence Formula

aₙ = a₁ + (n−1)d

a₁ is the first term, d is the common difference (added each step), and n is the term number you want.

The common difference d can be positive (increasing sequence), negative (decreasing sequence), or zero (constant sequence).

Recursive form: aₙ = aₙ₋₁ + d — each term is the previous term plus d. Both forms describe the same sequence.

To find d between any two consecutive terms, subtract: d = a₂ − a₁ = a₃ − a₂ = … The difference is always the same.

Linear Connection

aₙ = d·n + (a₁ − d)

Arithmetic sequences correspond exactly to linear functions. When you graph the terms (n, aₙ), the points always fall on a straight line.

The common difference d is the slope of that line. The value a₁ − d is the y-intercept (what you'd get at n = 0).

This is why the dot plot below always forms a straight line — arithmetic sequences are linear growth (or decay) in disguise.

Example: 3, 7, 11, 15… has d = 4, so slope = 4. Writing as a function: f(n) = 4n − 1. Check: f(1) = 3 ✓, f(2) = 7 ✓.