Calculate the nth term
aₙ = a₁ + (n - 1) × rSequence formulas
Arithmetic sequence
Calculate the nth term or the sum of the first n terms of an arithmetic sequence, or work backwards to the first term, the common difference, or the index.
Formula
aₙ = a₁ + (n - 1) × r; Sₙ = n × (a₁ + aₙ) / 2 In an arithmetic sequence every term differs from the previous one by the same value r, so the nth term grows linearly and the partial sum is the number of terms times the average of the first and the last term.
Arithmetic sequence calculator
Symbols and units
| Symbol | Quantity | SI | Supported units |
|---|---|---|---|
| a₁ | first termValue of the term with index 1, from which the whole sequence is generated. | 1 | 1 |
| r | common differenceConstant difference between two consecutive terms: r = aₙ₊₁ - aₙ. It may be negative or zero. | 1 | 1 |
| n | term indexPosition of the term in the sequence; a natural number greater than or equal to 1. | 1 | 1 |
| aₙ | nth termValue of the term with index n, calculated from the explicit formula. | 1 | 1 |
| Sₙ | sum of the first n termsSum a₁ + a₂ + ... + aₙ of the first n terms of the sequence. | 1 | 1 |
Formula rearrangements
Calculate the sum of n terms
Sₙ = n × (a₁ + aₙ) / 2Calculate the first term
a₁ = aₙ - (n - 1) × rCalculate the common difference
r = (aₙ - a₁) / (n - 1)Calculate the term index
n = (aₙ - a₁) / r + 1Worked examples
Twelfth term of a sequence
An arithmetic sequence starts with a₁ = 5 and has a common difference r = 3. Calculate a₁₂.
- 1Step 1: Use the explicit formula aₙ = a₁ + (n - 1) × r.
- 2Step 2: Substitute a₁ = 5, r = 3 and n = 12: a₁₂ = 5 + 11 × 3.
- 3Step 3: Multiply 11 × 3 = 33 and add 5 to obtain a₁₂ = 38.
Result: a₁₂ = 38
Sum of the first twenty terms
An arithmetic sequence has a₁ = 2 and r = 4. Calculate the sum of its first 20 terms.
- 1Step 1: First find the last term: a₂₀ = 2 + 19 × 4 = 78.
- 2Step 2: Use the sum formula Sₙ = n × (a₁ + aₙ) / 2.
- 3Step 3: Substitute: S₂₀ = 20 × (2 + 78) / 2 = 10 × 80 = 800.
Result: S₂₀ = 800 (a₂₀ = 78)
Sources and review
- Algebra and Trigonometry 2e, section 13.2: Arithmetic Sequences
OpenStax · Reviewed 29 August 2026
Definition of an arithmetic sequence and the explicit formula for its nth term.
- Algebra and Trigonometry 2e, section 13.4: Series and Their Notations
OpenStax · Reviewed 29 August 2026
Derivation of the sum of the first n terms of an arithmetic series.
- Wybrane wzory matematyczne na egzamin maturalny z matematyki
Centralna Komisja Egzaminacyjna · Reviewed 29 August 2026
Official Polish examination reference for arithmetic sequences and their sums.