An arithmetic sequence grows or shrinks by the same amount at every step: 3, 7, 11, 15, … adds 4 each time. These sequences describe evenly spaced seats, regular savings deposits without interest, stacked objects and any quantity that changes linearly. Enter the first term and common difference, or any two terms you know, to get the nth term, the sum of the first n terms, both formulas, a table of terms with running totals, and a check of whether a given number appears in the sequence.
How to use the arithmetic sequence calculator
- Choose what you know: the first term and common difference, or any two terms with their positions.
- Enter the term number n you want. The calculator also sums the first n terms.
- Optionally enter a number to test whether it belongs to the sequence.
- Read aₙ on the tape, along with the sum, the explicit and recursive formulas, and the table of terms.
Arithmetic sequence formulas
The recursive form is aₙ = aₙ₋₁ + d with a starting value a₁. From two known terms ap and aq:
Worked example
Take the sequence 3, 7, 11, 15, …, so a₁ = 3 and d = 4.
- Explicit formula: aₙ = 3 + (n − 1) × 4 = 4n − 1.
- The 10th term: a₁₀ = 3 + 9 × 4 = 39.
- The sum of the first 10 terms: S₁₀ = 10 × (3 + 39) ÷ 2 = 210.
- Is 39 a term? (39 − 3) ÷ 4 + 1 = 10, a whole number, so yes: it is the 10th. Is 40? (40 − 3) ÷ 4 + 1 = 10.25, so no.
Suppose instead you only know that the 3rd term is 11 and the 8th term is 31. Five steps separate them and the value rose by 20, so d = 20 ÷ 5 = 4 and a₁ = 11 − 2 × 4 = 3: the same sequence.
Where arithmetic sequences show up
The sum trick
The sum formula comes from pairing terms from both ends. Writing 1 + 2 + … + 100 forward and backward and adding gives 100 pairs that each total 101, so the sum is 100 × 101 ÷ 2 = 5,050. Every arithmetic series works the same way: n terms with an average of (a₁ + aₙ) ÷ 2.
Real-world patterns
- A theater whose first row has 20 seats, with 2 more seats in each following row, has 20 + 14 × 2 = 48 seats in row 15 and 15 × (20 + 48) ÷ 2 = 510 seats in the first 15 rows.
- Saving $50 more each month than the month before forms an arithmetic sequence of deposits; the total after a year is an arithmetic series.
- Straight-line depreciation removes the same amount each year, so book values form a decreasing arithmetic sequence.
Linear functions
Because aₙ = dn + (a₁ − d), an arithmetic sequence is a linear function evaluated at whole numbers, with slope d. If the differences between consecutive terms are not constant, the sequence is not arithmetic. Constant ratios point to a geometric sequence instead, and for sums of other patterns written in sigma notation, use the summation calculator.
Frequently asked questions
What is an arithmetic sequence?
A list of numbers in which each term is the previous term plus the same fixed amount, called the common difference d. Examples are 3, 7, 11, 15, … (d = 4) and 10, 7.5, 5, 2.5, … (d = −2.5). Graphed against position, the terms lie on a straight line.
How do I find the common difference from two terms?
Divide the change in value by the change in position: d = (a_q − a_p) ÷ (q − p). If the 3rd term is 11 and the 8th is 31, then d = (31 − 11) ÷ (8 − 3) = 4. The first term follows from a₁ = a_p − (p − 1)d = 11 − 2 × 4 = 3.
What is the formula for the sum of an arithmetic sequence?
Sₙ = n(a₁ + aₙ) ÷ 2, the number of terms times the average of the first and last term. An equivalent form that avoids finding aₙ first is Sₙ = n[2a₁ + (n − 1)d] ÷ 2.
How can I tell if a number belongs to the sequence?
Solve aₙ = x for n: n = (x − a₁) ÷ d + 1. The number is a term only if n is a positive whole number. In 3, 7, 11, …, the value 39 gives n = 10, so it is the 10th term; 40 gives n = 10.25, so it is not in the sequence.
What is the difference between a sequence and a series?
A sequence is the list of terms; a series is their sum. The arithmetic sequence 3, 7, 11, 15 corresponds to the arithmetic series 3 + 7 + 11 + 15 = 36.