Sequence Calculator
Sequence calculator — enter the first term, the common difference or ratio, and how many terms to get the nth term and the sum of an arithmetic or geometric sequence, plus a preview of the first terms. Runs in your browser.
Sequence Calculator
How to Use the Sequence Calculator
Pick the type
Arithmetic (adds) or geometric (multiplies).
Enter the first term
The value the sequence starts at.
Set the step
The common difference or ratio, and n.
Read the results
See the nth term, the sum and the first terms.
Two Patterns That Explain a Lot
Arithmetic and geometric sequences are the two simplest patterns in mathematics, and between them they describe a surprising amount of the world. An arithmetic sequence advances by adding a fixed amount each step — 2, 5, 8, 11 — while a geometric sequence advances by multiplying by a fixed amount — 3, 6, 12, 24. This calculator handles both: choose the type, enter the first term, the common difference or ratio, and the number of terms, and it returns the nth term, the sum of those terms, and a preview of the sequence so you can see the pattern unfold.
The formulas behind it are worth knowing because they recur throughout algebra and finance. The nth term of an arithmetic sequence is a + (n − 1)d, and its series sum is n/2 × (2a + (n − 1)d) — the elegant shortcut famously rediscovered by the young Gauss when he added the numbers 1 to 100 in seconds. For a geometric sequence the nth term is a × r^(n−1) and the sum is a × (r^n − 1) / (r − 1). The calculator picks the correct pair automatically and copes with negative and decimal inputs, so a negative common difference produces a decreasing sequence and a ratio between zero and one produces a decaying one.
These are not just classroom exercises. Compound interest is precisely a geometric sequence: each period the balance is multiplied by one plus the rate, so savings and loan growth follow the same nth-term and sum formulas. Depreciation, population models and many physical processes are geometric too, while anything that grows or shrinks by a steady fixed amount — a savings plan with equal monthly deposits, the seats added to each row of a theatre — is arithmetic. It is worth noting what these two patterns are not: the famous Fibonacci sequence, where each term is the sum of the previous two, is recursive and fits neither mould. For the two standard closed-form types, though, this tool gives exact answers and shows its working through the term preview — and as always, the whole calculation runs in your browser.
Compound interest is just a geometric sequence — which is why the same nth-term and sum formulas describe both homework and your savings.
10 Facts About Sequences
An arithmetic sequence adds a fixed difference each step.
A geometric sequence multiplies by a fixed ratio.
Arithmetic nth term: a + (n−1)d.
Geometric nth term: a × rⁿ⁻¹.
Arithmetic sum: n/2 × (2a + (n−1)d).
Geometric sum: a(rⁿ−1)/(r−1).
A geometric series with |r| < 1 has a finite infinite sum.
Compound interest is a geometric sequence.
The Fibonacci sequence is neither — it is recursive.
This calculator runs in your browser — nothing is uploaded.
Frequently Asked Questions
- An arithmetic sequence moves by adding a fixed amount — the common difference — each step, like 2, 5, 8, 11. A geometric sequence moves by multiplying by a fixed amount — the common ratio — each step, like 3, 6, 12, 24. This calculator handles both; you choose the type and it applies the matching formulas.
- For an arithmetic sequence the nth term is a + (n − 1)d, where a is the first term and d the common difference. For a geometric sequence it is a × r^(n−1), where r is the common ratio. So the 10th term of 2, 5, 8, … is 2 + 9 × 3 = 29.
- The arithmetic series sum of n terms is n/2 × (2a + (n − 1)d). The geometric series sum is a × (r^n − 1) / (r − 1), or simply a × n when the ratio is exactly 1. The calculator selects the right formula based on the sequence type you choose.
- The common difference is the fixed number added between consecutive terms of an arithmetic sequence (it can be negative for a decreasing sequence). The common ratio is the fixed number each term is multiplied by in a geometric sequence; a ratio between −1 and 1 makes the terms shrink toward zero.
- Yes. The first term, common difference and common ratio can all be negative or decimal. A negative common difference gives a decreasing arithmetic sequence; a ratio between 0 and 1 gives a decaying geometric one. The calculator handles all these cases and previews the first terms so you can see the behaviour.
- Yes — it lists the first several terms of the sequence so you can verify the pattern at a glance, in addition to computing the nth term and the running sum. Seeing the terms makes it easy to confirm you have entered the right type, first term and step.
- Yes, it is a geometric sequence: each period the balance is multiplied by one plus the interest rate. A balance growing at 5% per year follows a geometric sequence with ratio 1.05, so the nth-term and sum formulas here describe how savings and loans grow over time.
- Fibonacci (1, 1, 2, 3, 5, 8, …) is neither arithmetic nor geometric — each term is the sum of the two before it, which makes it a recursive sequence. This calculator covers the two standard closed-form sequence types; Fibonacci needs a different, recursive approach.
- The formulas are exact for the inputs you provide. Very large term counts with a large ratio can produce extremely big numbers that hit floating-point limits, but for normal educational and financial ranges the nth term and sum are precise.
- Completely free, with no account or limit. It works offline once the page has loaded and collects no data.
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