Find the next number in any sequence instantly. Identify arithmetic, geometric, and Fibonacci patterns with step-by-step formulas. Free online number sequence calculator for homework, puzzles, and pattern recognition.
In mathematics, a sequence is an ordered list of objects. Accordingly, a number sequence is an ordered list of numbers that follow a particular pattern. The individual elements in a sequence are often referred to as terms, and the number of terms in a sequence is called its length, which can be infinite. In a number sequence, the order is important, and depending on the sequence, it is possible for the same terms to appear multiple times.
Sequences have many applications in various mathematical disciplines due to their properties of convergence. A series is convergent if the sequence converges to some limit, while a sequence that does not converge is divergent. They are particularly useful as a basis for series, which are generally used in differential equations and mathematical analysis.
So you've got a list of numbers and you need to figure out what comes next. Maybe it's for homework. Maybe you're trying to solve a puzzle. Or maybe you just saw a pattern in some data and your brain won't let it go.
Whatever the reason, you're in the right place. Our Number Sequence Calculator does the heavy lifting for you. Just type in your numbers, hit calculate, and boom - you get the next term, the formula, and a step-by-step explanation.
But here's the thing. A calculator is great for getting answers fast. But understanding how sequences work? That's what's going to save you on tests and help you spot patterns in real life. Let's break it all down.
This is the simplest type. You add (or subtract) the same number every time. That number is called the common difference.
Example: 5, 10, 15, 20, 25 — Common difference? 5. You're adding 5 each time.
Example: 100, 90, 80, 70 — Common difference? -10. You're subtracting 10 each time.
The formula for the nth term of an arithmetic sequence is: an = a1 + (n - 1)d
Where a1 is the first term, n is the term number you want, and d is the common difference. Let's try: Sequence 3, 7, 11, 15... 10th term: a10 = 3 + (10-1)×4 = 3 + 36 = 39.
Instead of adding, you multiply (or divide) by the same number each time. That number is called the common ratio.
Example: 2, 4, 8, 16, 32 — Common ratio? 2. You're multiplying by 2 each time.
Example: 81, 27, 9, 3 — Common ratio? 1/3. You're dividing by 3.
The formula for the nth term of a geometric sequence is: an = a1 × r(n-1)
Let's try: 5, 10, 20, 40... 8th term: a8 = 5 × 27 = 5 × 128 = 640. See how fast geometric sequences grow?
This one's special. Each term is the sum of the two terms before it.
It starts: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... So 0+1=1, 1+1=2, 1+2=3, and so on.
The Fibonacci sequence shows up everywhere in nature. Pinecones, sunflowers, seashells, even the way your fingers are shaped. It's kind of wild.
But here's a trick most people don't know. The calculator doesn't just guess. It checks for arithmetic patterns first (adding), then geometric patterns (multiplying), then more complex ones like Fibonacci or quadratic sequences. If your sequence doesn't fit any standard pattern, the calculator will tell you. That's actually useful information - it means you might have a typo, or the sequence is following a rule that's too complex for a simple formula.
I've seen students make the same mistakes over and over. Here are the big ones:
Mistake 1: Assuming It's Arithmetic When It's Not
Just because the numbers go up doesn't mean you're adding the same amount each time. Check the differences between terms. If they're not the same, it's not arithmetic.
Mistake 2: Forgetting That Geometric Sequences Can Have Fractions
A common ratio can be a fraction. Like 1/2. So 100, 50, 25, 12.5 is a geometric sequence. Don't get thrown off by decimals.
Mistake 3: Mixing Up the Term Number
When using the formula, remember that the first term is n=1, not n=0. This trips up a lot of people.
Mistake 4: Not Checking Your Work
Always test your formula on the terms you already have. If your formula says the third term should be 15 but the actual third term is 12, something's wrong.
Sometimes you'll get a sequence that doesn't fit the standard types. Like: 1, 4, 9, 16, 25. This is actually the squares of 1, 2, 3, 4, 5. So the next term is 36 (6 squared).
Or: 1, 8, 27, 64 — These are cubes. Next term is 125 (5 cubed). Our calculator handles these too. But if you're doing it by hand, here's a tip: look at the differences between terms. If the differences themselves form a pattern, you might have a quadratic sequence.
For 1, 4, 9, 16, 25: Differences: 3, 5, 7, 9. Second differences: 2, 2, 2. When the second differences are constant, you're looking at a quadratic sequence (something with n²).
Always check if the difference between terms is constant first (arithmetic).
If not, check if the ratio between terms is constant (geometric).
If neither, look at the differences of the differences.
Write down the formula once you find it. Then test it on the first few terms.
Don't panic if you can't figure it out right away. Sometimes stepping away for a minute helps.
Most sequence calculators just spit out a number. Ours shows you the work. You'll see the formula, the common difference or ratio, and a step-by-step explanation.
That way, you're not just getting the answer. You're learning how to find it yourself next time.
And if you're stuck on a homework problem, you can use the calculator to check your work. Did you find the right pattern? The calculator will confirm it.
The Fibonacci sequence was actually known in India hundreds of years before Fibonacci wrote about it. Indian mathematicians used it to study poetry and music rhythms. They were figuring out how many different ways you could arrange long and short beats.
So next time someone says math isn't creative, tell them about the sequence that started with poetry.
First, look at the difference between each pair of numbers. If the difference is the same every time, you have an arithmetic sequence. Just add that difference to the last number. If the numbers are multiplying by the same factor, it's geometric. Multiply the last number by that factor. Our calculator does this automatically for you.
In an arithmetic sequence, you add or subtract the same number each time. Like 2, 4, 6, 8 (adding 2). In a geometric sequence, you multiply or divide by the same number each time. Like 2, 4, 8, 16 (multiplying by 2). Arithmetic is about constant addition. Geometric is about constant multiplication.
Just pick any two numbers that are next to each other and subtract the first from the second. So if your sequence is 10, 15, 20, 25, do 15 - 10 = 5. That's your common difference. Check it on another pair to make sure. 20 - 15 = 5. Yep, it's consistent.
Don't worry. Some sequences follow other rules. Look at the differences between the differences. If those are constant, you might have a quadratic sequence. Or it could be squares, cubes, or a famous sequence like Fibonacci. Our calculator checks for all these patterns automatically.
For arithmetic sequences, use an = a1 + (n - 1)d. Plug in the first term for a1, the common difference for d, and the term number you want for n. For geometric sequences, use an = a1 × r(n-1). The calculator shows you the formula for your specific sequence.
You might have a typo. Check that all numbers are separated by commas and there are no extra spaces or letters. Also make sure you have at least three numbers. The calculator needs enough data to figure out the pattern. If you only enter two numbers, there are too many possibilities.
Yes, absolutely. Decimal sequences work the same way as whole numbers. Just enter them with a decimal point. Like 1.5, 3.0, 4.5, 6.0. The calculator will find the pattern just as easily.
The Fibonacci sequence starts with 0 and 1. Then each number is the sum of the two before it. So it goes 0, 1, 1, 2, 3, 5, 8, 13... It's special because it shows up everywhere in nature. Pinecones, sunflowers, and even the spiral of a seashell follow this pattern. It's like nature's favorite math trick.
First, figure out the pattern using the numbers you have. Then use the formula to find the missing term. For example, if you have 2, _, 6, 8 and you know it's arithmetic with a common difference of 2, the missing number is 4. Our calculator can handle this if you leave a blank or use a question mark.