🔢 Arithmetic Sequence Calculator – Find Terms, Sums & Progressions
An arithmetic sequence (also called an arithmetic progression) is one of the most fundamental concepts in mathematics. Whether you are a student tackling algebra homework, a teacher preparing worked examples, or a professional modelling a linear growth pattern, this calculator solves every classic arithmetic-sequence problem in seconds — and shows the full working so you can learn from each step.
What is an Arithmetic Sequence?
An arithmetic sequence is an ordered list of numbers in which each term differs from the previous term by a constant amount called the common difference (d). For example:
3, 8, 13, 18, 23, … (d = 5)The first term is a₁, any term at position n is aₙ, and the constant step between consecutive terms is d.
Core Formulas Used
| What to Find | Formula |
|---|---|
| nth term | aₙ = a₁ + (n − 1)d |
| Common difference | d = (aⱼ − aᵢ) / (j − i) |
| First term | a₁ = aₙ − (n − 1)d |
| Sum of first n terms (Sₙ) | Sₙ = n/2 × (a₁ + aₙ) |
| Sum (alternative form) | Sₙ = n/2 × (2a₁ + (n − 1)d) |
Calculation Modes Explained
Find nth Term
Enter the first term (a₁), the common difference (d), and the term index (n). The calculator applies aₙ = a₁ + (n − 1)d and shows each substitution step. For example, with a₁ = 3, d = 5, n = 12: a₁₂ = 3 + 11 × 5 = 58.
Find Common Difference
If you know two terms at positions i and j, the calculator derives d = (aⱼ − aᵢ) / (j − i) and also back-calculates the first term so you can generate the full sequence.
Find First Term
Given a known term value at position k and the common difference, the calculator back-solves a₁ = aₙ − (n − 1)d so you can anchor the entire progression.
Sum of First n Terms
The arithmetic series sum uses the formula Sₙ = n/2 × (a₁ + aₙ). This is ideal for problems like "find the total salary paid over 12 months with a monthly raise of £200".
Generate Sequence Table
Enter a₁ and d, specify how many terms to display (up to 500), and receive an indexed table of term values alongside their cumulative sums (Sₙ). The table is exportable as a CSV file for use in spreadsheets.
Arithmetic Sequence vs Arithmetic Series
These two terms are often confused. An arithmetic sequence is the ordered list of terms themselves — e.g., 2, 5, 8, 11. An arithmetic series is the sumof those terms — e.g., 2 + 5 + 8 + 11 = 26. This tool handles both in a single interface.
Handling Negative and Decimal Differences
The calculator fully supports descending sequences (negative d) as well as non-integer common differences. For example, d = −7 produces 100, 93, 86, 79, … and d = 0.25 produces 1, 1.25, 1.5, 1.75, … — both equally valid arithmetic progressions.
Real-World Applications
- Finance: Savings plans with equal monthly contributions, salary increment schedules, loan amortisation patterns.
- Physics: Uniform acceleration — positions at equal time intervals under constant force.
- Construction: Equally spaced structural elements, staircase step heights.
- Education: Exam and homework problems covering sequences, series, and sigma notation.
Tips for Accurate Results
- Use the Precision setting (0–10 decimal places) to match your required accuracy.
- Negative common differences are entered directly — just type a minus sign (e.g.,
-3). - The Show Steps toggle reveals every substitution so you can verify or learn the method.
- Use Export CSV to download the full term table for analysis in Excel or Google Sheets.
- Example presets at the bottom of the tool let you explore common textbook problems instantly.