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Arithmetic Sequence Calculator | Compute Nth Term and Sum

Q: What is an arithmetic sequence?

The Arithmetic Sequence Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Finds the nth term and sum of any arithmetic sequence. Enter the first term, the common difference, and the position n you want, and the tool computes the nth term using the formula aₙ = a₁ + (n − 1)d, plus the sum of the first n terms, with the working There is no limit on usage, and it works on any device.

Q: What is the nth term formula?

The Arithmetic Sequence Calculator uses the standard aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ)/2. You enter your values, and the calculator applies the formula step by step, showing the working so you can verify the math and understand how the result is derived rather than trusting a black box.

Q: How do you find the sum of an arithmetic sequence?

Enter the required values into the input fields and press the calculate button. The tool applies the formula aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ)/2 and returns the result with a short explanation of each step, so you can check the working yourself.

Q: How do I find the common difference?

Subtract any term from the next one: d = aₙ₊₁ − aₙ. For 2, 5, 8, 11, the difference is 3. To confirm it is an arithmetic sequence, check that the gap is identical between several consecutive pairs. Once you have d and the first term, the formula aₙ = a₁ + (n − 1)d gives any term.

Q: Can the common difference be negative or a fraction?

Yes — a negative common difference makes the sequence decrease (10, 7, 4, 1…), and a fractional difference produces fractional terms (0.5, 1.0, 1.5, 2.0…). The formulas work identically for both, and the calculator handles them with full precision. A zero difference makes every term equal, which is still a valid arithmetic sequence.

Q: How do I find the common difference of a sequence?

Subtract any term from the one after it: for 7, 12, 17, 22 the common difference is 5. Check two or three gaps — if they differ, the sequence isn't arithmetic at all. A constant ratio instead of a constant gap (2, 6, 18, 54) means you're looking at a geometric sequence and need its tools.

Q: How do I find the nth term without listing every term?

Use the formula aₙ = a₁ + (n − 1)d directly: for a₁ = 2 and d = 3, the 50th term is 2 + 49 × 3 = 149 — no listing needed. The calculator takes the first term, difference, and n, and returns the term plus the sum of the first n terms, so long sequences are solved in one step.

Q: How do I get the 100th term without listing 100 numbers?

Use aₙ = a₁ + (n−1)d to jump straight there: with a₁ = 5 and d = 7, the 100th term is 5 + 99 × 7 = 698. Listing is never necessary — the formula exists precisely so you don't have to enumerate.

Q: Why does the sum formula average the first and last terms?

Because pairs symmetric around the middle always sum to the same value: in 5, 12, …, 138, the 1st + 20th = 143 and the 2nd + 19th = 143. So n × that average is the total — it's Gauss's pairing trick, and the page shows the pairing for your inputs.

Q: How do I find the common difference from two scattered terms?

Divide the value gap by the position gap: d = (aⱼ − aᵢ) ÷ (j − i). If the 3rd term is 19 and the 7th is 47, d = 28 ÷ 4 = 7. The page accepts any two known terms and derives d before computing anything else.

Find the nth term and the sum of the first n terms of any arithmetic sequence, with the formula and working shown. Enter the first term and common difference and get any term in the sequence instantly.

Last updated: July 2026 · Free · No sign-up required

Results

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Quick reference

Given	Formula	Example (a₁=3, d=5, n=10)
nth term	aₙ = a₁ + (n−1)d	a₁₀ = 3 + 9×5 = 48
Sum of n terms	Sₙ = n(a₁+aₙ)/2	S₁₀ = 10(3+48)/2 = 255
Common difference	d = a₂ − a₁	8 − 3 = 5

info Arithmetic Sequence Calculator

Free math calculator — enter your numbers and get an instant, accurate result.

info Private by design

Everything runs locally in your browser. No uploads, no accounts, no tracking.

info Works everywhere

Fully responsive and mobile-friendly — calculate on any device, any time.

info Educational

Includes the formula and step-by-step explanation so you understand the math, not just the answer.

Interpretation guide

Common difference d	Sequence behaviour
d > 0	Increasing (3, 8, 13, …)
d = 0	Constant — every term equals a₁
d < 0	Decreasing (10, 7, 4, …)

lightbulb Seats in the 20th row of a theater

A theater's rows grow arithmetically: the first row has 5 seats and each row adds 7 (d = 7). How many seats are in the 20th row, and what's the total capacity?

Result: 20th row: a₂₀ = 5 + 19 × 7 = 5 + 133 = 138 seats. Total: S₂₀ = 20 × (5 + 138) ÷ 2 = 10 × 143 = 1,430 seats.

What this means: The nth-term formula jumps straight to row 20 without listing 19 rows first, and the sum formula pairs rows (1st + 20th = 143, 2nd + 19th = 143…) into ten pairs. Real venues use exactly this math to quote capacity from a seating chart's first row and row-to-row growth.

Formula & methodology

Formula: aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ)/2

How arithmetic sequences are solved

An arithmetic sequence adds a constant difference d to each term. Two formulas cover everything: the nth term aₙ = a₁ + (n−1)d jumps directly to any position, and the sum Sₙ = n(a₁ + aₙ) ÷ 2 totals the first n terms via Gauss's pairing — first + last, second + second-to-last, every pair identical.

Take the first term a₁, the common difference d, and the position n.
Compute aₙ = a₁ + (n−1)d.
Compute Sₙ = n × (a₁ + aₙ) ÷ 2.

Example: a₁ = 5, d = 7, n = 20 → 20th term 138, sum 1,430. If you know two scattered terms instead of d, use d = (aⱼ − aᵢ) ÷ (j − i) first.

Authoritative source: Wolfram MathWorld

Frequently asked questions

What is an arithmetic sequence?

The Arithmetic Sequence Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Finds the nth term and sum of any arithmetic sequence. Enter the first term, the common difference, and the position n you want, and the tool computes the nth term using the formula aₙ = a₁ + (n − 1)d, plus the sum of the first n terms, with the working There is no limit on usage, and it works on any device.

What is the nth term formula?

The Arithmetic Sequence Calculator uses the standard aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ)/2. You enter your values, and the calculator applies the formula step by step, showing the working so you can verify the math and understand how the result is derived rather than trusting a black box.

How do you find the sum of an arithmetic sequence?

Enter the required values into the input fields and press the calculate button. The tool applies the formula aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ)/2 and returns the result with a short explanation of each step, so you can check the working yourself.

How do I find the common difference?

Subtract any term from the next one: d = aₙ₊₁ − aₙ. For 2, 5, 8, 11, the difference is 3. To confirm it is an arithmetic sequence, check that the gap is identical between several consecutive pairs. Once you have d and the first term, the formula aₙ = a₁ + (n − 1)d gives any term.

Can the common difference be negative or a fraction?

Yes — a negative common difference makes the sequence decrease (10, 7, 4, 1…), and a fractional difference produces fractional terms (0.5, 1.0, 1.5, 2.0…). The formulas work identically for both, and the calculator handles them with full precision. A zero difference makes every term equal, which is still a valid arithmetic sequence.

How do I find the common difference of a sequence?

Subtract any term from the one after it: for 7, 12, 17, 22 the common difference is 5. Check two or three gaps — if they differ, the sequence isn't arithmetic at all. A constant ratio instead of a constant gap (2, 6, 18, 54) means you're looking at a geometric sequence and need its tools.

How do I find the nth term without listing every term?

Use the formula aₙ = a₁ + (n − 1)d directly: for a₁ = 2 and d = 3, the 50th term is 2 + 49 × 3 = 149 — no listing needed. The calculator takes the first term, difference, and n, and returns the term plus the sum of the first n terms, so long sequences are solved in one step.

How do I get the 100th term without listing 100 numbers?

Use aₙ = a₁ + (n−1)d to jump straight there: with a₁ = 5 and d = 7, the 100th term is 5 + 99 × 7 = 698. Listing is never necessary — the formula exists precisely so you don't have to enumerate.

Why does the sum formula average the first and last terms?

Because pairs symmetric around the middle always sum to the same value: in 5, 12, …, 138, the 1st + 20th = 143 and the 2nd + 19th = 143. So n × that average is the total — it's Gauss's pairing trick, and the page shows the pairing for your inputs.

How do I find the common difference from two scattered terms?

Divide the value gap by the position gap: d = (aⱼ − aᵢ) ÷ (j − i). If the 3rd term is 19 and the 7th is 47, d = 28 ÷ 4 = 7. The page accepts any two known terms and derives d before computing anything else.

Does it work for decreasing or fractional sequences?

Yes — a negative d gives a decreasing sequence (10, 7, 4, …) and a fractional d gives fractional terms (0.5, 1.0, 1.5, …). The formulas are exact for all real values; the behavior table on the page shows which regime your difference falls into.

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