What is my arithmetic sequence?
Type the first term, the common difference, and n. The arithmetic sequence calculator finds the nth term and the sum of the first n terms, lists the first terms, and writes the explicit and recursive formulas.
- nth term (aₙ)
- 29
Term 10 is 29, and the sum of terms 1 to 10 is 155.
- Sum of the first n terms (Sₙ)
- 155
- First terms
- 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
- Explicit formula
- a_n = 2 + 3(n - 1)
- Recursive formula
- a_1 = 2, a_n = a_(n - 1) + 3
nth term (aₙ): 29. Term 10 is 29, and the sum of terms 1 to 10 is 155.
How to calculate
Finds the nth term and the sum of the first n terms of an arithmetic sequence from its first term and common difference, with the first terms and the explicit and recursive formulas.
Example with the default inputs (First term (a₁) 2, Common difference (d) 3, Term number (n) 10): Term 10 is 29, and the sum of terms 1 to 10 is 155.
Method: aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ)/2.
- Terms are numbered from 1; the first term is a₁.
- The numbers are read as the exact decimals typed and the formulas are worked out in exact fractions; each result is then the nearest 64-bit float.
- A term or sum beyond about 1.8 × 10^308 has no answer.
Worked examples
Each example is checked against the calculator on every build.
- First term (a₁) 2, Common difference (d) 10, Term number (n) 5 gives nth term (aₙ) 42, Sum of the first n terms (Sₙ) 110, First terms 2, 12, 22, 32, 42, Explicit formula a_n = 2 + 10(n - 1).Source: OpenStax, Algebra and Trigonometry 2e, section 13.2 Arithmetic Sequences (aₙ = a₁ + d(n − 1)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-2-arithmetic-sequences, retrieved 2026-10-02, example: 2, 12, 22, 32, 42 has aₙ = 2 + 10(n − 1) and a₅ = 42
- First term (a₁) 5, Common difference (d) 3, Term number (n) 10 gives nth term (aₙ) 32, Sum of the first n terms (Sₙ) 185, Recursive formula a_1 = 5, a_n = a_(n - 1) + 3.Source: OpenStax, Algebra and Trigonometry 2e, section 13.4 Series and Their Notations (Sₙ = n(a₁ + aₙ)/2), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-4-series-and-their-notations, retrieved 2026-10-02, example: 5 + 8 + … + 32 = 185
- First term (a₁) 8, Common difference (d) 2, Term number (n) 5 gives nth term (aₙ) 16, Sum of the first n terms (Sₙ) 60.Source: OpenStax, Algebra and Trigonometry 2e, section 13.2 Arithmetic Sequences (aₙ = a₁ + d(n − 1)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-2-arithmetic-sequences, retrieved 2026-10-02, example: a₁ = 8, d = 2 gives a₅ = 16
- First term (a₁) 10, Common difference (d) -4, Term number (n) 6 gives nth term (aₙ) -10, Sum of the first n terms (Sₙ) 0, Explicit formula a_n = 10 - 4(n - 1), First terms 10, 6, 2, -2, -6, -10.
- First term (a₁) 0.1, Common difference (d) 0.2, Term number (n) 3 gives nth term (aₙ) 0.5, Sum of the first n terms (Sₙ) 0.9, First terms 0.1, 0.3, 0.5.
How it works
This page is the arithmetic setting of the sequence calculator, with the same inputs, rules and outputs. Terms are numbered from 1. With first term a₁, common difference d, and term number n:
- nth term: aₙ = a₁ + (n − 1)d
- sum of the first n terms: Sₙ = n(a₁ + aₙ)/2
The numbers are read as the exact decimals typed (0.1 is exactly one tenth), the formulas are worked out in exact fractions, and each result is then the nearest 64-bit floating-point number, shown to at most 6 decimal places. A term or sum beyond the largest 64-bit float (about 1.8 × 10^308) has no answer: "The nth term is too large to show." or "The sum is too large to show." The first term and the difference are from −10^9 to 10^9, and n is a whole number from 1 to 10,000.
The calculator also shows:
- First terms: a₁, a₂, … up to 10 terms (n terms when n is less than 10). Each is written as a decimal in full (no exponent, no thousands separators, a plain hyphen for a minus sign) when it needs at most 20 decimal places and at most 30 digits, and otherwise as the shortest text of its nearest 64-bit float in JavaScript’s number notation (
1e+25). They are joined by a comma and a space, followed by, …when n is more than 10. - Explicit formula:
a_n = a₁ + d(n - 1)with a₁ and d written as the terms are;- |d|(n - 1)for a negative d,+ (n - 1)for d = 1,- (n - 1)for d = −1, anda_n = a₁for d = 0. - Recursive formula:
a_1 = a₁, a_n = a_(n - 1) + d, or- |d|for a negative d.
Worked examples by hand
a₁ = 2, d = 10, n = 5. 2, 12, 22, 32, 42: a₅ = 2 + 4 × 10 = 42, S₅ = 5 × (2 + 42)/2 = 110; explicit formula a_n = 2 + 10(n - 1).
a₁ = 5, d = 3, n = 10. a₁₀ = 5 + 9 × 3 = 32, S₁₀ = 10 × (5 + 32)/2 = 185.
a₁ = 8, d = 2, n = 5. a₅ = 8 + 4 × 2 = 16, S₅ = 5 × (8 + 16)/2 = 60.
a₁ = 10, d = −4, n = 6. 10, 6, 2, −2, −6, −10, which add up to 0; explicit formula a_n = 10 - 4(n - 1).
a₁ = 0.1, d = 0.2, n = 3. 0.1, 0.3, 0.5, and 0.1 + 0.3 + 0.5 = 0.9 exactly.
Other questions people ask
What is an arithmetic sequence?
A list of numbers where the same amount, the common difference d, is added each time. 2, 12, 22, 32, 42 adds 10 each time, so d = 10. A negative d makes the terms fall: 10, 6, 2, −2.
How do I find the nth term?
Use aₙ = a₁ + (n − 1)d. For 2, 12, 22, …, the 5th term is 2 + 4 × 10 = 42. The formula simplifies to aₙ = 10n − 8.
How do I find the common difference?
Subtract any term from the next one: 12 − 2 = 10. If you know two terms that are not next to each other, divide their difference by the gap in positions: with a₁ = 8 and a₄ = 14, d = (14 − 8) ÷ 3 = 2.
How do I add up an arithmetic sequence?
Multiply the number of terms by the average of the first and last terms: Sₙ = n(a₁ + aₙ)/2. For 5 + 8 + 11 + … + 32 (10 terms): 10 × (5 + 32) ÷ 2 = 185.
What is the difference between an arithmetic sequence and an arithmetic series?
The sequence is the list of terms (5, 8, 11, …); the series is their sum (5 + 8 + 11 + …). The calculator gives both: the terms and the sum of the first n.
What is the recursive formula?
The first term and the rule for each next term: a₁ = 5, aₙ = aₙ₋₁ + 3. The explicit formula, aₙ = 5 + 3(n − 1), gives any term straight from n.