Arithmetic Sequence
Calculate the nth term, sum of n terms, and full sequence of any arithmetic progression instantly. Free online arithmetic sequence calculator with step-by-step results.
About This Calculator
An arithmetic sequence (also called an arithmetic progression) is a sequence of numbers in which the difference between any two consecutive terms is always the same. This constant difference is known as the common difference, denoted as d. For instance, the sequence 3, 7, 11, 15, 19 is arithmetic because each term increases by 4.
Our arithmetic sequence calculator computes three key results instantly. The nth term (aₙ) is calculated using the formula aₙ = a1 + (n-1)d. The sum of the first n terms (Sₙ) is given by Sₙ = n/2 x (2a1 + (n-1)d). The calculator also displays the full sequence as a comma-separated list so you can inspect every term individually.
This calculator works for any real-number inputs, including positive, negative, decimal, and fractional values. Whether you are a student studying algebra, a teacher preparing lesson examples, or a professional working with financial projections, this tool provides accurate results instantly. The calculator supports large values for the number of terms, limited only by your browser's memory.
Regional Notes
Arithmetic sequences are taught universally in mathematics curricula. In India, they appear in CBSE Class 10 and 11 mathematics under progressions. In the US, arithmetic sequences are covered in Algebra I and II courses. In the UK, they are part of the GCSE and A-Level mathematics specifications. The formulas used are standard across all educational systems worldwide.
Frequently Asked Questions
What is an arithmetic sequence?
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This fixed difference is called the common difference (d). For example, 2, 5, 8, 11, 14 is an arithmetic sequence with first term 2 and common difference 3.
How do you find the nth term of an arithmetic sequence?
The nth term of an arithmetic sequence is given by the formula aₙ = a1 + (n-1)d, where a1 is the first term, d is the common difference, and n is the term position. Simply enter these values into the calculator to get the nth term instantly.
What is the sum of an arithmetic series formula?
The sum of the first n terms of an arithmetic series is Sₙ = n/2 x (2a1 + (n-1)d), which can also be written as Sₙ = n/2 x (a1 + aₙ). This formula works for any arithmetic progression and is derived from pairing terms from both ends.
How is the arithmetic sequence calculator useful for students?
Students can use this calculator to verify homework problems, check their work when finding nth terms or sums, and visualize how arithmetic sequences grow. It supports any real numbers for the first term and common difference, including decimals and negative values.
Can the common difference be negative?
Yes, the common difference can be negative, zero, or positive. A negative common difference produces a decreasing sequence (e.g., 10, 7, 4, 1, -2). A zero common difference results in a constant sequence where every term equals the first term.
What real-world applications use arithmetic sequences?
Arithmetic sequences appear in many real-world scenarios: calculating monthly loan payments with fixed principal reduction, predicting seating arrangements in auditoriums, determining salary increments with fixed annual raises, and computing total distance covered during steady acceleration.