Use the calculator below to list every term from your first value a₁, common difference d, and term count n. Labeled results and a step-by-step solution update instantly in your browser with no sign-up.
Each term follows aₖ = a₁ + (k - 1)d. Enter a₁, d, and n; the list and solution update as you type.
Result
Enter a valid first term a₁, common difference d, and a whole number of terms n (1 to 50) to see the list.
Using this calculator
An arithmetic sequence is an ordered list of numbers where each term after the first is found by adding the same fixed amount, called the common difference d. If you know the first term a₁ and d, every later term is predictable.
These patterns appear in algebra, pre-calculus, and everyday planning. Steady pay raises, equal spacing on a number line, and simple linear trends all follow the same add-d-each-time rule.
nth term: aₙ = a₁ + (n - 1)d
a₁ = first term | d = common difference | n = term position (1, 2, 3, …)
The formula breaks into three parts: the starting value a₁, the total change from adding d repeatedly, and the position n. The factor (n - 1) counts how many times d is added after the first term.
Variable guide: treat n as the index label (first term is n = 1). If a problem gives the count of terms to list, apply the formula for each position k from 1 through that count.
You can list terms by repeated addition or by evaluating aₙ = a₁ + (n - 1)d at each position. Both methods should agree when your inputs are consistent.
Worked examples mirror the calculator: enter the same a₁, d, and n to compare your hand work with the tool at the top of this page.
A sequence starts at 2 and increases by 5 each step.
Result: a₁ = 2, a₂ = 7, a₃ = 12, a₄ = 17, a₅ = 22
A decreasing pattern drops 3 each step from 20.
Result: a₁ = 20, a₂ = 17, a₃ = 14, a₄ = 11
A half-step increase from 1.
Result: a₁ = 1, a₂ = 1.5, a₃ = 2, a₄ = 2.5, a₅ = 3, a₆ = 3.5
Quarterly savings of $25 starting at $100 (simple model, no interest).
Result: a₁ = 100, a₂ = 125, a₃ = 150, a₄ = 175
Sₙ = n/2 (a₁ + aₙ)
Alternate: Sₙ = n/2 [2a₁ + (n - 1)d]
An arithmetic series adds the terms of an arithmetic sequence. The first-plus-last method pairs endpoints: multiply n/2 by (a₁ + aₙ).
Practical usage: list terms with the calculator, identify aₙ, then apply Sₙ. Example: 2, 7, 12, 17, 22 has a₁ = 2, a₅ = 22, n = 5, so S₅ = 5/2 × 24 = 60.
Formula comparison: use aₙ = a₁ + (n - 1)d to find the last term when only a₁ and d are given, then plug into Sₙ = n/2 (a₁ + aₙ).
The sequence answers what each term is; the series answers what the total is when you add them. Keeping the labels separate prevents half-correct homework.
| Topic | Sequence | Series |
|---|---|---|
| Definition | Ordered list of terms | Sum of those terms |
| Formula focus | aₙ = a₁ + (n - 1)d | Sₙ = n/2 (a₁ + aₙ) |
| Typical question | What is the 8th term? | What is the total of the first 8 terms? |
| Common confusion | Stopping after listing terms | Forgetting to add after listing |
d = a₂ - a₁
Also d = a₃ - a₂ = aₙ - aₙ₋₁ for consecutive terms
Subtract neighbors in the same order every time. Increasing sequences have positive d; decreasing sequences have negative d.
Error checking: if differences vary, the list is not arithmetic. Identify the pattern before forcing this formula.
Fixed-step situations map to a₁ and d once you name the starting value and the repeat amount.
Equal payment plans and simple savings schedules (without compounding) follow arithmetic steps.
Walking the same extra blocks each day or adding one cup of water per hour creates predictable lists.
Cutting fabric into sections that each grow by a fixed length is an arithmetic length sequence.
Algebra and exam prep use these lists to practice notation, graphs, and sums.
Tables pair index k with value aₖ. Graphs plot those pairs and form a straight line when d is constant.
Slope interpretation: the slope equals d. The intercept reflects a₁ adjusted for how the axis is drawn.
Visual patterns: equal spacing in k should show equal vertical change when d is fixed.
The interactive tool stays at the top of this page so you can calculate first, then read the explanations below.
Enter first term a₁, common difference d, and number of terms n. The result panel lists a₁ through aₙ with subscripts, and the step-by-step block walks through the formula.
Example check: a₁ = 2, d = 3, n = 4 should give 2, 5, 8, 11. Try those values in the tool and match the written example section.
Most errors are sign slips, index confusion, or mixing sequences with series.
Arithmetic adds d each step (linear list). Geometric multiplies by ratio r each step (curved growth). Pick the model that matches equal differences vs equal ratios.
aₙ = a₁ + (n - 1)d. The calculator evaluates this for each position from 1 through your chosen term count.
The sequence is the list of terms. The series is their sum, found with Sₙ = n/2 (a₁ + aₙ) when you need a total.
Yes. Negative d decreases terms. Fractional d creates decimal steps. Enter either form directly in the calculator.
You can list up to 50 terms (whole number n). For longer lists, use the formula on paper or a spreadsheet.
No. Calculations run locally in your browser with no account required.