Arithmetic Sequence Calculator

What is an arithmetic sequence calculator?

An arithmetic sequence calculator finds the nth term and the running total of an arithmetic sequence from just three inputs: the first term, the common difference, and how many terms you want. Enter the first term a₁, the common difference d, and a term number n, press Calculate, and the tool returns the value of the nth term (aₙ) and the sum of the first n terms (Sₙ). It also previews the opening terms of the sequence, up to the first 12, so you can see the pattern at a glance.

An arithmetic sequence (also called an arithmetic progression) is a list of numbers in which each term differs from the previous one by the same fixed amount. That fixed amount is the common difference. The sequence 3, 8, 13, 18, 23, … is arithmetic because every step adds 5. This calculator does the two jobs that take the most arithmetic by hand — jumping straight to a distant term, and adding a long run of terms — without you writing out the whole list.

How does the calculator work?

The tool uses the two standard closed-form formulas for arithmetic progressions:

• nth term: aₙ = a₁ + (n − 1)d
• Sum of the first n terms: Sₙ = n/2 × (2a₁ + (n − 1)d)

Here is what each symbol means:

• a₁ — the first term, where the sequence starts. It can be any real number, positive or negative.
• d — the common difference, the constant amount added from one term to the next. A positive d makes the sequence increase, a negative d makes it decrease, and a d of 0 makes every term equal to a₁.
• n — the term number (the position in the list). It is a whole counting number: n = 1 is the first term, n = 2 the second, and so on.
• aₙ — the value of the term at position n.
• Sₙ — the total you get by adding the first n terms together (the arithmetic series).

The nth-term formula works because reaching the nth term means starting at a₁ and taking n − 1 steps of size d. The sum formula is the classic pairing trick: the average of the first and last term, (a₁ + aₙ) / 2, multiplied by the number of terms n. Since aₙ = a₁ + (n − 1)d, that average expands to (2a₁ + (n − 1)d) / 2, which gives Sₙ = n/2 × (2a₁ + (n − 1)d). The units of aₙ and Sₙ are simply whatever units a₁ and d are in — the formulas are pure arithmetic and carry no units of their own.

Examples

Each example below matches the calculator exactly. Type the same inputs into the tool above to reproduce every number.

Example 1: a₁ = 3, d = 5, n = 10

• nth term: a₁₀ = a₁ + (n − 1)d = 3 + (10 − 1) × 5 = 3 + 9 × 5 = 3 + 45 = 48
• Sum: S₁₀ = n/2 × (2a₁ + (n − 1)d) = 10/2 × (2×3 + 9×5) = 5 × (6 + 45) = 5 × 51 = 255
• Sequence preview: 3, 8, 13, 18, 23, …

Example 2: a₁ = 2, d = 3, n = 5

• nth term: a₅ = 2 + (5 − 1) × 3 = 2 + 4 × 3 = 2 + 12 = 14
• Sum: S₅ = 5/2 × (2×2 + 4×3) = 2.5 × (4 + 12) = 2.5 × 16 = 40
• Sequence preview: 2, 5, 8, 11, 14

Example 3: a₁ = 10, d = −2, n = 6 (a decreasing sequence)

• nth term: a₆ = 10 + (6 − 1) × (−2) = 10 + 5 × (−2) = 10 − 10 = 0
• Sum: S₆ = 6/2 × (2×10 + 5×(−2)) = 3 × (20 − 10) = 3 × 10 = 30
• Sequence preview: 10, 8, 6, 4, 2, 0

A negative common difference simply means the sequence counts down, and both formulas handle it without any change.

nth term and sum reference table

This table is built from Example 1 (a₁ = 3, d = 5). It shows how each term aₙ and the running sum Sₙ grow as the position n increases.

n	aₙ = a₁ + (n−1)d	Sₙ = n/2 × (2a₁ + (n−1)d)
1	3	3
2	8	11
3	13	24
4	18	42
5	23	65
6	28	93
10	48	255

Notice that each aₙ rises by the common difference of 5, while each Sₙ is the previous sum plus the new term — a quick way to sanity-check the closed-form answers.

What is an arithmetic sequence used for?

Arithmetic progressions show up wherever a quantity changes by a fixed step:

• Savings and payments. Setting aside the same amount each month, or paying down a loan by an equal principal each period, forms an arithmetic sequence; Sₙ is the running total.
• Salary and pay raises. A fixed annual raise (for example, the same dollar increment each year) makes your yearly pay an arithmetic sequence, and the sum is your cumulative earnings.
• Seating, stairs and stacking. Rows of seats that grow by a constant number, steps that rise a fixed height, or stacked items in a triangular pile are all arithmetic patterns.
• Schoolwork and exams. nth-term and series questions are core topics in algebra, GCSE and SAT maths, and this tool shows the working so you can check each step.
• Physics with constant change. Distances covered in equal time intervals under constant acceleration follow an arithmetic pattern.

Tips and common mistakes

• Use n − 1, not n, in the term formula. The most frequent error is writing a₁ + nd. Reaching the nth term takes only n − 1 steps, so the 10th term of 3, 5, … uses 9 steps, giving 3 + 9d.
• Find d by subtracting consecutive terms. The common difference is any term minus the one before it: in 3, 8, 13 it is 8 − 3 = 5. Make sure the gap is truly constant before treating a list as arithmetic.
• Don’t confuse the sequence with the series. aₙ is a single term’s value; Sₙ is the sum of all terms up to that point. This tool reports both, so read the right label.
• Keep the sign of a negative difference. With d = −2 the sequence decreases; plug −2 directly into both formulas rather than subtracting twice.
• n must be a whole number. Term positions are counting numbers (1, 2, 3, …); there is no “2.5th term” in an arithmetic sequence.

Limitations and notes

This calculator covers arithmetic sequences only — those with a constant difference between terms. It does not handle geometric sequences, where each term is multiplied by a constant ratio (such as 2, 6, 18, 54). The term preview is capped at the first 12 terms to keep the display readable, but aₙ and Sₙ are computed exactly from the formulas for any valid n. Results follow standard floating-point arithmetic, so very large term numbers or non-integer inputs may show tiny rounding artefacts at the last decimal place. Term positions are assumed to start at n = 1, which is the usual convention; if a problem indexes from n = 0, shift your value of n by one before entering it.

For more number work, try the scientific notation calculator, the percentage calculator and the ratio calculator in the full math calculators category.