Sequences and Series Calculator
Find the nth term and the sum of the first n terms of an arithmetic or geometric sequence, and preview the sequence itself.
Solve Your Sequence
Choose your sequence type, enter the first term and common difference or ratio, and how many terms you want to sum.
Sequence Solver
Works out the nth term and the sum of the first n terms.
How it Works
Understanding sequences and series
Identify the Pattern
An arithmetic sequence adds a fixed common difference to get the next term; a geometric sequence multiplies by a fixed common ratio instead.
Apply the nth Term Formula
Arithmetic: aₙ = a₁ + (n − 1)d. Geometric: aₙ = a₁ × r^(n − 1), using your chosen term number.
Sum the Series
We add up the first n terms using the matching sum formula, rather than adding each term one by one.
Check for Convergence
For a geometric sequence with a common ratio between -1 and 1, we also show the sum of the infinite series, since it settles on a fixed value.
Sequences & Series FAQ
Answers to the most frequently asked questions about arithmetic and geometric sequences and series.
A sequence is an ordered list of numbers, such as 2, 4, 6, 8. A series is what you get when you add the terms of a sequence together, such as 2 + 4 + 6 + 8. A calculator that finds ‘the sum’ is working with a series, while one that finds ‘the nth term’ is working with the underlying sequence.
In an arithmetic sequence, each term is found by adding a fixed number, called the common difference, to the previous term. In a geometric sequence, each term is found by multiplying the previous term by a fixed number, called the common ratio. This difference changes both how terms are calculated and how the sequence grows over time.
For an arithmetic sequence, the nth term is the first term plus the common difference multiplied by one less than the term number. For a geometric sequence, the nth term is the first term multiplied by the common ratio raised to the power of one less than the term number.
An infinite geometric series only has a finite sum if the common ratio is strictly between -1 and 1. In that case, the sum equals the first term divided by one minus the common ratio. If the common ratio is 1, -1, or has a size greater than 1, the series either doesn’t settle on a value or grows without bound.
A negative common ratio makes the terms of a geometric sequence alternate between positive and negative values. A common ratio between -1 and 1, such as a fraction, makes each term smaller in size than the last, so the sequence shrinks toward zero as it progresses.