Number Sequence Calculator

Enter your known values in the calculator above to find the next number in a sequence, compute a specific nth term, or identify the pattern type. This free number sequence calculator handles arithmetic, geometric, and Fibonacci sequences with step-by-step solutions.

Whether you need the 50th term of an arithmetic progression or want to confirm a geometric pattern, the tool does the math instantly. Below you'll find how each sequence type works, the formulas behind the results, and guidance on picking the right solver for your problem.

How the Number Sequence Calculator Works Step by Step

1. Choose a sequence type. Select arithmetic, geometric, or Fibonacci from the calculator options.
2. Enter your known values. Depending on the type, you'll input a first term, a common difference or common factor, or a set of existing numbers.
3. Specify what you need. Pick whether you want the next number in the sequence, a specific nth term, or a partial sum.
4. Read the results. The calculator returns the answer along with a step-by-step breakdown of the formula it used.

No sign-up required. The tool runs entirely in your browser.

If you already have a list of numbers and aren't sure which pattern they follow, enter the first several terms. The calculator will attempt to identify the sequence type and confirm or reject common patterns.

Find the Next Number in a Sequence

The most common reason people reach for a sequence calculator is simple: "What comes next?"

To find the next number in the sequence, the tool looks at the differences (or ratios) between consecutive terms. Here's what it checks:

• Constant difference between terms points to an arithmetic sequence.
• Constant ratio between terms points to a geometric sequence.
• Each term equals the sum of the two previous terms points to a Fibonacci sequence.

For example, given 3, 7, 11, 15, the difference between each pair is 4. The next number is 19.

Given 2, 6, 18, 54, each term is multiplied by 3. The next number is 162.

Some number patterns don't fit neatly into one category. The limitations section below covers when automatic detection may not work.

Arithmetic Sequence Calculator and Solver

An arithmetic sequence is a list of numbers where the difference between consecutive terms stays the same. That fixed gap is called the common difference (often labeled d).

Examples you'll recognize:

• 5, 10, 15, 20 (common difference = 5)
• 100, 93, 86, 79 (common difference = −7)

This is the most commonly encountered sequence type in algebra courses, standardized tests, and everyday planning problems like evenly spaced payments or scheduling intervals.

Use the Arithmetic Sequence Calculator to Find Any Term

To use the arithmetic sequence calculator, you need two pieces of information:

1. First term (a₁): The starting value of the sequence.
2. Common difference (d): The amount added to each term to get the next one.

Then specify which term you want. The calculator returns that term's value and shows the work. You can also request a partial sum of the first n terms.

Quick example: First term = 4, common difference = 6, find the 25th term. Result: a₂₅ = 4 + (25 − 1) × 6 = 148.

Formula for Arithmetic Sequences

The formula for the nth term of an arithmetic sequence:

aₙ = a₁ + (n − 1) × d

Where:

• aₙ = the value of the term at position n
• a₁ = the first term
• n = the position number you want
• d = the common difference

To find the sum of the first n terms:

Sₙ = n / 2 × (a₁ + aₙ)

This sum formula is useful when you need a cumulative total, like adding up a series of evenly increasing deposits.

Geometric Sequence Calculator and Solver

A geometric sequence (also called a geometric progression) is a list of numbers where each term is found by multiplying the previous term by a fixed value. That multiplier is called the common ratio (often labeled r).

Examples:

• 3, 12, 48, 192 (common ratio = 4)
• 1000, 500, 250, 125 (common ratio = 0.5)

Geometric sequences show up in compound interest, population growth models, and signal processing. Any time a value grows or shrinks by a percentage, you're looking at a geometric pattern.

Use the Geometric Sequence Calculator to Find Any Term

Enter these inputs:

1. First term (a₁): The starting value.
2. Common ratio (r): The factor each term is multiplied by.

Then choose the term position. The calculator solves for that value and displays each step.

Quick example: First term = 5, common ratio = 3, find the 8th term. Result: a₈ = 5 × 3⁷ = 5 × 2187 = 10,935.

Formula for Geometric Sequences

The formula for the nth term of a geometric sequence:

aₙ = a₁ × r⁽ⁿ⁻¹⁾

Where:

• aₙ = the value at position n
• a₁ = the first term
• r = the common ratio
• n = the position number

To find the sum of the first n terms (when r ≠ 1):

Sₙ = a₁ × (1 − rⁿ) / (1 − r)

When the absolute value of r is less than 1, the series converges and you can also calculate an infinite sum: S∞ = a₁ / (1 − r).

Fibonacci Sequences: Find the Next Number

A Fibonacci sequence is a sequence where each term equals the sum of the two previous terms. The classic version starts with 0 and 1:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34 …

The pattern is straightforward: pick any two consecutive numbers, add them, and you get the next one.

Fibonacci sequences appear in nature (spiral patterns in shells and sunflowers), computer science (algorithm analysis), and mathematical modeling. They don't have a single common difference or ratio, which is why they need their own approach.

How to calculate a Fibonacci sequence: The recursive formula is simply Fₙ = Fₙ₋₁ + Fₙ₋₂. You start with two seed values (typically F₀ = 0 and F₁ = 1) and build from there.

There is also a closed-form expression called Binet's formula, but for practical purposes the recursive method is easier to follow and is what the calculator uses internally.

Enter your two starting numbers in the tool, specify how many terms you need, and the calculator generates the full list.

When to Use Each Sequence Calculator Tool

The three main sequence types this tool covers are arithmetic, geometric, and Fibonacci. Matching your list to the right type is the fastest path to the next term or nth term.

Choosing the right solver depends on the pattern in your numbers.

Practical guidance:

• Use the arithmetic sequence calculator for evenly spaced values: salary raises by a fixed dollar amount, seating rows that add a constant number of chairs, or test score progressions.
• Use the geometric sequence calculator for percentage-based growth or decay: investment returns, depreciation, bacterial doubling, or halving problems.
• Use the Fibonacci solver when each value clearly depends on the two values before it.

If your numbers don't match any of these types of number sequences, the pattern may involve a quadratic, factorial, or combination rule that a basic sequence calculator cannot identify. See limitations below.

Limitations of a Number Sequence Solver

This tool gives estimates and pattern matches based on the formulas above. Keep a few things in mind:

• Ambiguous short lists. A sequence of only two or three numbers can match multiple patterns. Provide at least four terms for more reliable detection.
• Non-standard sequences. The calculator covers arithmetic, geometric, and Fibonacci sequences. It does not solve for polynomial sequences of higher order, look-and-say sequences, or sequences defined by custom recursive equations.
• Floating-point precision. Very large term positions or extreme ratios may produce rounding in the display. The underlying mathematics is correct, but displayed decimals may be truncated.
• Not a proof. Identifying a pattern from a finite list of terms does not mathematically prove the rule continues. The tool finds the most likely match.
• Educational use. Results are a planning and learning aid. For academic submissions, verify the formula and show your work independently.

If your sequence doesn't fit neatly into these three types, consider whether a combination of patterns or a different mathematical model applies. For specialized or research-level series analysis, consult a mathematics reference or instructor.

Step-by-Step Solutions: How to Calculate an Arithmetic Sequence

Working through the formula by hand builds understanding. Here's the process.

1. Identify your first term (a₁).
2. Determine the common difference (d) by subtracting any term from the next.
3. Decide which term position (n) you need.
4. Plug values into aₙ = a₁ + (n − 1) × d.
5. If you also need the sum, use Sₙ = n / 2 × (a₁ + aₙ).

Step-by-Step Example With a Number Sequence

Problem: Find the 8th term and the sum of the first 8 terms of the sequence: 5, 9, 13, 17, …

1. First term: a₁ = 5
2. Common difference: d = 9 − 5 = 4
3. Find the 8th term: a₈ = 5 + (8 − 1) × 4 = 5 + 28 = 33
4. Find the sum: S₈ = 8 / 2 × (5 + 33) = 4 × 38 = 152

The 8th term is 33, and the sum of the first 8 terms is 152. The calculator above performs these same steps automatically.

Common Difference in an Arithmetic Progression

The common difference (d) is the engine of every arithmetic sequence. It determines whether the sequence rises, falls, or stays flat.

• Positive d: The sequence increases (3, 7, 11, 15 …).
• Negative d: The sequence decreases (20, 14, 8, 2 …).
• Zero d: Every term is the same (5, 5, 5, 5 …).

How do you find the common difference? Subtract any term from the term directly after it. d = a₂ − a₁.

Take the sequence: −12, −1, 10, 21. The common difference is −1 − (−12) = 11. You can verify: 10 − (−1) = 11 and 21 − 10 = 11. Consistent result, so d = 11.

If you know two non-adjacent terms, you can still solve for d. Use: d = (aₙ − aₘ) / (n − m), where m and n are the two term positions.

Step-by-Step Solutions for Every Calculation

This number sequence calculator doesn't just hand you an answer. It shows the work.

For each result, you'll see:

• The type of sequence detected (arithmetic, geometric, Fibonacci, or other)
• The common difference, common ratio, or rule identified
• The formula applied
• The calculated next term

Step-by-step solutions help with math homework, exam prep, and building your own pattern recognition skills. If you're studying sequences and series, reading through the solution reinforces the concepts behind the answer.

You can also use the step-by-step output to verify your own manual calculations. Enter a sequence you've already solved by hand, then compare your steps to the calculator's breakdown.

Sequence Calculator Math: Formulas Behind the Solver

Here's a quick reference of the core formulas this calculator uses:

Arithmetic sequence (nth term): aₙ = a₁ + (n − 1) × d

Arithmetic sequence (sum of first n terms): Sₙ = n/2 × (a₁ + aₙ)

Geometric sequence (nth term): aₙ = a₁ × r⁽ⁿ⁻¹⁾

Geometric sequence (sum of first n terms, r ≠ 1): Sₙ = a₁ × (1 − rⁿ) / (1 − r)

Fibonacci-type sequence: aₙ = aₙ₋₁ + aₙ₋₂

These are the explicit formulas behind every calculation. The solver applies whichever formula matches the detected pattern. Understanding these equations helps you interpret results and apply them to more advanced mathematical problems.

Number Sequence Calculator FAQs

What is an arithmetic sequence calculator used for?

An arithmetic sequence calculator finds any term (or the next number) when the difference between consecutive terms stays constant. Enter the first term and the common difference, then choose the position you need. It is the right tool for evenly spaced patterns like 5, 10, 15, 20.

What are the 3 types of sequences?

The three sequence types most people need are:

• Arithmetic — add a fixed common difference each step.
• Geometric — multiply by a fixed common ratio each step.
• Fibonacci — each term is the sum of the two terms before it.

This number sequence calculator solves all three and shows the formula used.

How do you calculate a Fibonacci sequence?

Start with two seed values (usually 0 and 1). Each new term equals the sum of the previous two: Fₙ = Fₙ₋₁ + Fₙ₋₂. Enter those seeds in the tool and request how many terms you need; the calculator builds the list with step-by-step notation.

How do I find the next number in a sequence?

Enter at least four known terms when you can. The solver checks for a constant difference (arithmetic), a constant ratio (geometric), or a Fibonacci-style rule, then returns the next term and explains which pattern matched.