Arithmetic Sequence Calculator
Use this arithmetic sequence calculator to find the nth term, common difference, and sum of a sequence. Supports step-by-step formulas, examples, and sequence preview.
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Arithmetic Sequence Calculator
Compute the nth term and the sum of an arithmetic sequence from the first term, common difference, and number of terms. Includes live sequence preview and formula walkthrough.
Interactive arithmetic sequence calculator
d can be positive, negative, or fractional (e.g. 2.5).
n must be a positive whole number (1, 2, 3, …).
Key results
- nth term (aₙ)
- –
- Sum of first n terms (Sₙ)
- –
Sequence preview
First 10 terms (or fewer if n < 10):
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What is an arithmetic sequence?
An arithmetic sequence (or arithmetic progression) is a list of numbers where the difference between any two consecutive terms is constant. This constant value is called the common difference and is usually denoted by d.
General form of an arithmetic sequence
a₁, a₂, a₃, …, aₙ, …
a₂ = a₁ + d
a₃ = a₁ + 2d
…
aₙ = a₁ + (n − 1)d
Formula for the nth term
If you know the first term \(a_1\), the common difference \(d\), and the position \(n\), you can find the nth term with:
aₙ = a₁ + (n − 1)d
This is simply adding the common difference \(d\) a total of \(n-1\) times to the first term.
Formula for the sum of the first n terms
The sum of the first n terms of an arithmetic sequence is denoted by \(S_n\). There are two common forms, depending on the information you have:
Using first term and common difference
Sₙ = n/2 × [2a₁ + (n − 1)d]
Using first and nth term
Sₙ = n/2 × (a₁ + aₙ)
Both formulas are equivalent. The second is often easier if you already know \(a_n\).
Worked example
Suppose you have an arithmetic sequence with:
- First term \(a_1 = 3\)
- Common difference \(d = 5\)
- Number of terms \(n = 10\)
The nth term is:
a₁₀ = 3 + (10 − 1) × 5 = 3 + 9 × 5 = 3 + 45 = 48
The sum of the first 10 terms is:
S₁₀ = 10/2 × [2 × 3 + (10 − 1) × 5]
= 5 × [6 + 9 × 5]
= 5 × (6 + 45) = 5 × 51 = 255
Recognising arithmetic sequences in practice
Arithmetic sequences appear in many everyday and professional contexts, for example:
- Regular salary increases (e.g. +€150 per month each year).
- Inventory counts when you add or remove the same number of units in each time period.
- Linear patterns in statistics or basic modelling, where a quantity changes by a fixed amount per step.
Arithmetic sequence FAQs
Formula (LaTeX) + variables + units
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General form of an arithmetic sequence a₁, a₂, a₃, …, aₙ, … a₂ = a₁ + d a₃ = a₁ + 2d … aₙ = a₁ + (n − 1)d
Using first term and common difference Sₙ = n/2 × [2a₁ + (n − 1)d] Using first and nth term Sₙ = n/2 × (a₁ + aₙ)
a₁₀ = 3 + (10 − 1) × 5 = 3 + 9 × 5 = 3 + 45 = 48
S₁₀ = 10/2 × [2 × 3 + (10 − 1) × 5] = 5 × [6 + 9 × 5] = 5 × (6 + 45) = 5 × 51 = 255
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- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
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https://consumer.ftc.gov/
Last code update: 2026-01-19
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