Arithmetic Sequence Calculator
Free Arithmetic sequence Calculator for sequences. Enter values to get step-by-step solutions with formulas and graphs.
Calculator
Adjust values & calculateSequence Terms
Partial Sums
Formula
Where an is the nth term, a1 is the first term, d is the common difference, n is the number of terms, and Sn is the sum of the first n terms. The nth term formula finds any specific term, while the sum formula calculates the total of all terms from the first through the nth.
Last reviewed: December 2025
Worked Examples
Example 1: Stadium Seating Layout
Example 2: Salary with Annual Raises
Background & Theory
The Arithmetic Sequence Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Arithmetic Sequence Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
an = a1 + (n-1)d | Sn = (n/2)(2a1 + (n-1)d)
Where an is the nth term, a1 is the first term, d is the common difference, n is the number of terms, and Sn is the sum of the first n terms. The nth term formula finds any specific term, while the sum formula calculates the total of all terms from the first through the nth.
Worked Examples
Example 1: Stadium Seating Layout
Problem: A stadium has 20 rows. The first row has 15 seats and each subsequent row has 3 more seats. Find the number of seats in the 20th row and the total seats.
Solution: a1 = 15, d = 3, n = 20\n20th row: a20 = 15 + (20-1) x 3 = 15 + 57 = 72 seats\nTotal seats: S20 = (20/2) x (15 + 72) = 10 x 87 = 870\nArithmetic mean = 870 / 20 = 43.5 seats per row
Result: 20th row: 72 seats | Total: 870 seats | Mean: 43.5 per row
Example 2: Salary with Annual Raises
Problem: An employee starts at $45,000 and receives a $2,500 raise each year. What is the salary in year 10 and total earnings over 10 years?
Solution: a1 = 45000, d = 2500, n = 10\nYear 10 salary: a10 = 45000 + (10-1) x 2500 = 45000 + 22500 = $67,500\nTotal earnings: S10 = (10/2) x (45000 + 67500) = 5 x 112500 = $562,500\nAverage salary = $562,500 / 10 = $56,250
Result: Year 10 salary: $67,500 | Total 10-year earnings: $562,500
Frequently Asked Questions
What is an arithmetic sequence and what defines it?
An arithmetic sequence (also called an arithmetic progression) is an ordered list of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by d. For example, 2, 5, 8, 11, 14 is an arithmetic sequence with first term a1 = 2 and common difference d = 3. Each term equals the previous term plus d. Arithmetic sequences appear naturally in everyday scenarios like counting by fives, monthly salary increments, or evenly spaced fence posts. They are one of the most fundamental concepts in mathematics and serve as building blocks for more advanced topics.
What is the formula for the nth term of an arithmetic sequence?
The nth term formula is: an = a1 + (n - 1) x d, where a1 is the first term, d is the common difference, and n is the position number. This formula lets you find any term without listing all preceding terms. For example, in the sequence 7, 12, 17, 22, ... the 100th term is 7 + (100 - 1) x 5 = 7 + 495 = 502. The formula can also work backwards: if you know a term value and want its position, rearrange to n = ((an - a1) / d) + 1. This is useful for determining whether a specific number belongs to a given arithmetic sequence.
How do I calculate the sum of an arithmetic sequence?
The sum of the first n terms uses the formula: Sn = (n / 2) x (2a1 + (n - 1) x d), or equivalently Sn = (n / 2) x (a1 + an) where an is the last term. This formula was famously discovered by young Carl Friedrich Gauss when asked to sum the numbers 1 through 100. He recognized that pairing the first and last terms (1 + 100 = 101, 2 + 99 = 101, etc.) creates 50 pairs of 101, giving 5,050. The formula generalizes this pairing technique to any arithmetic sequence. It is one of the most elegant and practical formulas in elementary mathematics.
How are arithmetic sequences used in real-life applications?
Arithmetic sequences model situations with constant rates of change. Linear depreciation uses them when an asset loses the same dollar amount each year: a $50,000 machine depreciating $5,000 annually follows the sequence 50000, 45000, 40000, and so on. Salary schedules with fixed annual raises form arithmetic sequences. Stacking objects in rows where each row has one more item than the previous creates arithmetic sequences. Seating arrangements in theaters (rows getting wider by a fixed number of seats), mortgage amortization with fixed principal payments, and drug dosage accumulation at regular intervals all involve arithmetic progressions.
What is the arithmetic mean and how does it relate to the sequence?
The arithmetic mean of an arithmetic sequence equals the average of the first and last terms: Mean = (a1 + an) / 2. This also equals the middle term if the number of terms is odd. For the sequence 5, 8, 11, 14, 17, the arithmetic mean is (5 + 17) / 2 = 11, which is indeed the middle (third) term. The arithmetic mean has an important property: any term in an arithmetic sequence is the arithmetic mean of its two neighbors. That is, an = (a(n-1) + a(n+1)) / 2. This property provides a quick way to verify that a sequence is arithmetic and to find missing terms within the sequence.
How do I find missing terms in an arithmetic sequence?
To find missing terms, first determine the common difference from any two known consecutive terms or from any two terms and their positions. If you know a3 = 10 and a7 = 26, then d = (26 - 10) / (7 - 3) = 4. Then use a1 = a3 - 2d = 10 - 8 = 2 to find the first term. Now fill in all missing terms: a4 = 14, a5 = 18, a6 = 22. If you have three terms and need to determine which value makes them arithmetic, use the property that the middle term equals the average of the outer two terms. This technique is commonly tested in standardized math examinations and competition problems.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy