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Sequence Calculator

Quick answer

This calculator computes and analyze mathematical sequences online.

  • Free, no sign-up, works on any device.
  • Instant results with a clear explanation of each step.

Figures are estimates from this calculator’s standard formula — adjust the inputs in the tool below for your exact number.

How the Sequence Calculator Works

Arithmetic: an = a1 + (n-1)d | Geometric: an = a1 x r^(n-1) | Fibonacci: Fn = Fn-1 + Fn-2

A sequence is an ordered list of numbers following a specific pattern. Arithmetic sequences add a constant difference, geometric sequences multiply by a constant ratio, and Fibonacci sequences sum the two previous terms.

  1. 1

    Select Sequence Type

    Choose arithmetic, geometric, or Fibonacci sequence

  2. 2

    Enter First Term

    Input the starting value of the sequence

  3. 3

    Enter Parameter

    For arithmetic, enter common difference; for geometric, enter common ratio

  4. 4

    Specify Position

    Enter the term position and number of terms to display

  5. 5

    View Results

    See the nth term value, sum, and sequence preview

Use Cases

Financial Planning

Calculate loan payments, savings growth, and compound interest

Population Modeling

Model exponential growth of populations using geometric sequences

Physics Problems

Calculate distance traveled under uniform acceleration

Algorithm Analysis

Understand recursive patterns and time complexity in programming

Tips

  • 1

    Arithmetic sequence: constant difference between consecutive terms

  • 2

    Geometric sequence: constant ratio between consecutive terms

  • 3

    Sum of arithmetic sequence: Sn = n/2 x (a1 + an)

  • 4

    Infinite geometric sum exists only when |r| < 1

Common Mistakes

  • Confusing arithmetic and geometric sequences - one adds, the other multiplies

  • Using n instead of (n-1) in formulas - the first term has no difference or ratio applied

  • Forgetting Fibonacci standard start values - usually begins with 0, 1 or 1, 1

  • Calculating infinite sum when |r| >= 1 - series only converges if |r| < 1

FAQs

What is the difference between arithmetic and geometric sequences?

Arithmetic adds a constant (like 2, 5, 8, 11 with d=3). Geometric multiplies by a constant (like 2, 6, 18, 54 with r=3).

How do I find the nth term of an arithmetic sequence?

Use an = a1 + (n-1)d. For sequence 3, 7, 11..., a10 = 3 + (10-1)x4 = 39.

What is the Fibonacci sequence?

Each term is the sum of the two preceding terms: 0, 1, 1, 2, 3, 5, 8, 13, 21... Found in nature, art, and architecture.

How do I calculate the sum of an arithmetic sequence?

Sn = n/2 x (a1 + an) or Sn = n/2 x (2a1 + (n-1)d).

When does an infinite geometric sum exist?

Only when |r| < 1. The sum is S = a1/(1-r). Example: 1 + 1/2 + 1/4 + ... = 2.

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