Sequence Calculator Online — Free Sequence Calculator Tool
CodexFuse Team · 2026-11-05T15:56:08.473Z · 10 min read min read
Find the nth term and sum of any arithmetic or geometric sequence in seconds — free, no sign-up, with a worked example.
A sequence calculator finds the nth term and the sum of an arithmetic or geometric sequence from just three inputs: the first term, the common difference or ratio, and the position n. You enter the values once and get the exact term, the total, and the first ten terms — no sign-up, no spreadsheet, no arithmetic slip at step seven.
Sequences turn up in more places than most people expect: a loan balance shrinking by a fixed amount each month, a mailing list growing by the same percentage, or marks across a test series. The maths is simple in principle and surprisingly easy to get wrong under time pressure, which is exactly when a dedicated tool earns its place.
What Is a Sequence Calculator?
A sequence calculator is a free online tool that computes the terms of a number sequence from its starting value and its pattern. Give it the first term a1, the common difference d for an arithmetic sequence or the common ratio r for a geometric one, and the position n, and it returns the nth term, the sum of the first n terms, and the first ten terms of the sequence.
That last part matters more than it sounds. Seeing the first ten terms lets you check the pattern with your own eyes before you trust the answer. If the terms do not match the sequence you had in mind, you know an input is wrong before the mistake reaches your homework or worksheet.
The CodexFuse sequence calculator handles both sequence types in one place:
- Arithmetic sequences: each term adds the same difference d, such as 2, 5, 8, 11, where d is 3.
- Geometric sequences: each term multiplies by the same ratio r, such as 3, 6, 12, 24, where r is 2.
- nth term: the value at any position n, from 1 to 1,000.
- Sum: the total of the first n terms, worked out with the standard sum formula.
- First ten terms: the sequence written out so you can verify the pattern at a glance.
How Do You Find the nth Term of an Arithmetic Sequence?
For an arithmetic sequence, the nth term is an = a1 + (n - 1)d. Take 2, 5, 8, ... where a1 is 2 and d is 3: the 10th term is 2 + 9 × 3 = 29. The pattern is a fixed step added each time, so each term grows in a straight line as n increases.
The formula works because you start at a1 and take (n - 1) steps of size d to reach the nth term. For the 10th term you take 9 steps, not 10 — the single most common off-by-one error in sequence problems. A calculator removes that trap entirely.
Before you trust any nth-term answer, check the difference between the first two terms yourself. If 2, 5, 8 gives d = 3 but you typed 4, every later term is wrong — and the error grows with n.
How Do You Find the nth Term of a Geometric Sequence?
For a geometric sequence, the nth term is an = a1 × r^(n - 1). Take 3, 6, 12, ... where a1 is 3 and r is 2: the 8th term is 3 × 2^7 = 384. Here each step multiplies rather than adds, so the terms grow far faster than in an arithmetic sequence.
That speed is why hand calculation goes wrong. By the 8th term you are multiplying by 128, and by the 20th term by over half a million. One missed power and the answer is off by a factor of two. The calculator handles the exponent exactly, and the first-ten-terms list shows the doubling (or halving) so you can see the ratio is right.
| Sequence type | Rule | nth term formula | Example |
|---|---|---|---|
| Arithmetic | Add d each step | an = a1 + (n - 1)d | 2, 5, 8, ... 10th term = 29 |
| Geometric | Multiply by r each step | an = a1 × r^(n - 1) | 3, 6, 12, ... 8th term = 384 |
| Constant (r = 1) | No change | an = a1, sum = a1 × n | 5, 5, 5, ... 20th term = 5 |
A Worked Example: The Worksheet That Had to Be Ready by 9 a.m.
Amna is a maths tutor in Lahore. On Sunday evening she is preparing a worksheet on sequences for a group of ninth-grade students, and she wants two clean problems with verified answers — one arithmetic, one geometric — plus the full working for her answer key.
Her first problem uses the sequence 3, 7, 11, 15, ... She needs the 20th term and the sum of the first 20 terms. Working it by hand means 19 steps of adding 4, then a sum formula with two chances to slip a digit. Instead, she opens the sequence calculator and does this:
- Choose arithmetic: the sequence adds the same amount each time, so she picks the arithmetic option.
- Enter a1 = 3, d = 4, n = 20: the first term is 3, the step is 4, and she wants the 20th term.
- Read the results: the 20th term is 79, and the sum of the first 20 terms is 820.
She checks the working before printing: 20th term = 3 + 19 × 4 = 79, and sum = 20/2 × (2 × 3 + 19 × 4) = 10 × 82 = 820. Both match, and the first ten terms listed by the tool — 3, 7, 11, 15, 19, 23, 27, 31, 35, 39 — confirm the pattern is the one she intended.
Her second problem is geometric: 100, 50, 25, ... She switches the tool to geometric, enters a1 = 100, r = 0.5, n = 6, and gets the 6th term 3.125 and the sum 196.875 — verified as 100 × 0.5^5 = 3.125. Both problems, including verification, take under two minutes, and the worksheet goes to print with an answer key she trusts.
Enter a1, d or r, and n in the free CodexFuse sequence calculator and get the nth term, the sum, and the first ten terms in seconds — then get back to the work that actually needs your brain.
Get Started with CodexFuseHow Do You Find the Sum of a Sequence?
The sum of the first n terms of an arithmetic sequence is Sn = n/2 × (2a1 + (n - 1)d). For a geometric sequence it is Sn = a1 × (r^n - 1) / (r - 1). As a check, the first 10 terms of 2, 5, 8, ... sum to 155, and the first 8 terms of 3, 6, 12, ... sum to 765.
The most famous sum of all is 1 to 100, which totals 5,050 — fifty pairs of 101, the trick the young Gauss is said to have spotted. The arithmetic sum formula is the same idea written once and reused forever: average the first and last terms, then multiply by how many terms there are.
If you work with percentages as often as sequences, the percentage difference calculator guide shows how to compare two numbers without falling into the baseline trap.
What Are the Limits of a Sequence Calculator?
Honest limits matter, because a tool used outside its range gives confident wrong answers. The CodexFuse sequence calculator handles n from 1 to 1,000, and a1 and d up to 1,000,000 in magnitude. Results that overflow show “Result too large” rather than a wrong number — which is the correct behaviour, but it means very large geometric sequences (a ratio of 2 past roughly n = 60 with a large starting term) will not resolve.
It also only handles the two classic types. It does not solve recurrence relations such as the Fibonacci sequence, where each term depends on the previous two, and it does not find a pattern from a list of numbers you paste in — you must know whether your sequence is arithmetic or geometric and supply a1, d or r.
When r = 1, the geometric sequence is constant: every term equals a1 and the sum is simply a1 × n. If your results look flat, check whether you entered a ratio of 1 by mistake.
If your problem is a series — the sum written out with plus signs, or a question about convergence — the companion series calculator is the right tool. A sequence is the list of terms; a series is their sum. Confusing the two is the most common conceptual error in this topic, and using the wrong tool for the question guarantees a wrong answer.
When Should You Use a Sequence Calculator Instead of Working by Hand?
Use the calculator whenever n is large, the numbers are awkward, or the answer feeds into something else. A 20th term with a decimal ratio or a sum of 500 terms are cases where a single slip costs more time to find than the calculation ever saved.
Work by hand when you are learning the concept or checking a result. Doing the first three terms manually, then comparing with the tool's first-ten-terms list, is the fastest way to catch a wrong input.
Students preparing for board exams get a double benefit: the tool checks their practice answers instantly, and the worked formulas reinforce the method. The FSc marks calculator guide covers the same check-your-work approach for exam totals, and a maths quiz generator can turn the same sequence problems into timed practice.
Frequently Asked Questions
How do I find the nth term of an arithmetic sequence?
Use an = a1 + (n - 1)d. For 2, 5, 8, ... with a1 = 2 and d = 3, the 10th term is 2 + 9 × 3 = 29. Remember to multiply by (n - 1), not n — you take nine steps to reach the tenth term.
How do I find the nth term of a geometric sequence?
Use an = a1 × r^(n - 1). For 3, 6, 12, ... with a1 = 3 and r = 2, the 8th term is 3 × 2^7 = 384. The exponent is (n - 1) for the same reason: seven multiplications take you from the first term to the eighth.
What is the sum of 1 to 100?
5,050. Pair the numbers from opposite ends — 1 + 100, 2 + 99, and so on — to get fifty pairs of 101. The arithmetic sum formula gives the same result: 100/2 × (2 + 99) = 5,050.
Does the calculator show the sequence itself?
Yes. It lists the first ten terms alongside the nth term and the sum, so you can verify the pattern at a glance. For the preset 100, 50, 25, ... the 6th term is 3.125 and the sum of the first six terms is 196.875.
What is the difference between a sequence and a series?
A sequence is the ordered list of terms — 2, 5, 8, 11. A series is the sum of those terms — 2 + 5 + 8 + 11. The sequence calculator finds terms and their total; for series-specific questions such as convergence, use the CodexFuse series calculator instead.
Is the sequence calculator free?
Yes. It is free to use with no sign-up and no limits on how many sequences you calculate. If you run a tutoring site or study blog, you can host it on CodexFuse — one managed platform on credits, with no subscriptions and no server babysitting.
Stop Doing Sequence Maths by Hand
Open the free sequence calculator now, enter a1, d or r, and n, and get the nth term, the sum, and the first ten terms in seconds. Then move the rest of your workflow to the same place: host your tutoring site or study blog on CodexFuse, where hosting, business email, security, and backups are fully managed on pay-as-you-go credits.
