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Arithmetic Sequence Calculator

Calculate the nth term and sum of arithmetic sequences with step-by-step solutions, interactive visualizations, and high-precision results up to 1000 decimal places.

Arithmetic Sequence Calculator
Quick Examples
📐 Basic (2,5,8...) 📉 Decreasing 🔬 Decimals ➖ Negative Start

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About Arithmetic Sequence Calculator

Welcome to the Arithmetic Sequence Calculator, a professional-grade tool for computing the nth term and sum of arithmetic sequences with high precision. Whether you're a student learning sequences, a teacher preparing materials, or a professional working with mathematical series, this calculator provides accurate results with step-by-step explanations and visual representations.

What is an Arithmetic Sequence?

An arithmetic sequence (also called an arithmetic progression or AP) is a sequence of numbers where each term after the first is obtained by adding a constant value called the common difference to the preceding term. This creates a linear pattern that increases, decreases, or remains constant depending on the common difference.

For example, the sequence 2, 5, 8, 11, 14, ... is an arithmetic sequence with:

Key Formulas

The nth Term Formula

To find any term in an arithmetic sequence, use this formula:

nth Term of Arithmetic Sequence
$$a_n = a_1 + (n - 1)d$$

Where:

Sum of Arithmetic Sequence

To calculate the sum of the first n terms, use one of these equivalent formulas:

Sum of First n Terms
$$S_n = \frac{n(a_1 + a_n)}{2} = \frac{n[2a_1 + (n-1)d]}{2}$$

The first form is useful when you know both the first and last terms. The second form is useful when you only know the first term and common difference.

How to Use This Calculator

  1. Enter the first term (a₁): Input the starting value of your sequence. This can be any real number, including decimals and negative values.
  2. Enter the common difference (d): Input the constant value added between terms. Positive values create increasing sequences; negative values create decreasing sequences.
  3. Enter n: Specify which term you want to find and how many terms to sum.
  4. Select precision: Choose the number of decimal places for calculations (10 to 1000).
  5. Calculate: Click the button to see the nth term, sum, sequence preview, visualization, and step-by-step solution.

Understanding Your Results

Types of Arithmetic Sequences

Type Common Difference Example Pattern
Increasing d > 0 3, 7, 11, 15, 19 Terms grow larger
Decreasing d < 0 20, 15, 10, 5, 0 Terms grow smaller
Constant d = 0 5, 5, 5, 5, 5 All terms equal

Real-World Applications

Finance & Economics

Science & Engineering

Everyday Examples

Arithmetic vs Geometric Sequences

Property Arithmetic Sequence Geometric Sequence
Pattern Add constant difference Multiply by constant ratio
nth Term aₙ = a₁ + (n-1)d aₙ = a₁ × rⁿ⁻¹
Graph Shape Linear (straight line) Exponential (curve)
Example 2, 5, 8, 11, 14 2, 6, 18, 54, 162

Frequently Asked Questions

What is an arithmetic sequence?

An arithmetic sequence (or arithmetic progression) is a sequence of numbers where each term after the first is obtained by adding a constant value called the common difference (d) to the previous term. For example, 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3.

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

Use the formula aₙ = a₁ + (n-1)d, where a₁ is the first term, n is the position, and d is the common difference. For example, to find the 10th term of sequence 3, 7, 11, ...: a₁₀ = 3 + (10-1)×4 = 3 + 36 = 39.

How do you calculate the sum of an arithmetic sequence?

Use Sₙ = n(a₁ + aₙ)/2 or Sₙ = n[2a₁ + (n-1)d]/2. The first formula requires knowing the first and last terms; the second only needs the first term and common difference.

What is the common difference?

The common difference (d) is the constant value added to each term to get the next term. Calculate it by subtracting any term from the next: d = a₂ - a₁. It can be positive, negative, or zero.

Can arithmetic sequences have negative numbers?

Yes. The first term can be negative, the common difference can be negative (decreasing sequence), or both. Example: -10, -7, -4, -1, 2 has first term -10 and common difference 3.

Additional Resources

Reference this content, page, or tool as:

"Arithmetic Sequence Calculator" at https://MiniWebtool.com/arithmetic-sequence-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 30, 2026

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

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