Nth Term Calculator - Sequence Calculator

What is an Arithmetic Sequence and Geometric Sequence?

An arithmetic sequence is defined by a constant difference between consecutive terms, while a geometric sequence is built by a common ratio. Understanding these patterns is the foundation for using any sequence calculator effectively. The nth term calculator handles both types, giving you immediate access to terms far beyond the first few.

Many students first encounter arithmetic sequence examples in algebra, such as 3, 7, 11, 15, where the difference is 4. Geometric sequence examples include 2, 6, 18, 54, where the ratio is 3. This tool automatically detects the pattern from your inputs, so you don't have to memorize complex rules.

How to Use the Sequence Calculator

Using this nth term calculator is straightforward. First, choose the sequence type from the dropdown: arithmetic or geometric. Next, enter the first term (a₁), the common difference (d) or common ratio (r), and the number of terms (n). Click "Calculate Sequence" and the tool will instantly generate the full sequence, the nth term, and the sum of the first n terms.

The arithmetic sequence formula used is aₙ = a₁ + (n-1)d. The geometric sequence formula is aₙ = a₁ × rⁿ⁻¹. For sums, the arithmetic sequence sum formula is Sₙ = n/2 × [2a₁ + (n-1)d], while the geometric sum is Sₙ = a₁ × (rⁿ - 1)/(r - 1) when r ≠ 1. All calculations are shown step-by-step, so you can follow the logic and learn how to find the nth term yourself.

Arithmetic Sequence Formula and Geometric Sequence Formula Knowledge

Mastering the arithmetic sequence formula and geometric sequence formula opens doors to real-world problem solving. In finance, an arithmetic sequence can model linear savings growth, while a geometric sequence models compound interest. In science, geometric sequences describe population growth or radioactive decay. In technology, arithmetic sequences help design evenly spaced UI elements, and geometric sequences are used in typographic scales and game difficulty curves.

The sum of arithmetic sequence is particularly useful for calculating totals over time, such as total savings after regular deposits. By understanding these formulas, you can predict outcomes without manually listing every term—a task this sequence calculator performs instantly for any n you choose.

What is an arithmetic sequence?

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant is called the common difference (d). For example, 2, 5, 8, 11 is an arithmetic sequence with d = 3. You can quickly find any term using the nth term calculator without listing all previous terms.

How to find the nth term?

To find the nth term, use the appropriate formula. For arithmetic sequences, apply aₙ = a₁ + (n-1)d. For geometric sequences, apply aₙ = a₁ × rⁿ⁻¹. Enter your values into the sequence calculator above, and it will compute the nth term instantly, showing all steps so you can learn the process.

What is the sum of arithmetic sequence formula?

The sum of the first n terms of an arithmetic sequence is given by Sₙ = n/2 × [2a₁ + (n-1)d]. This formula calculates the total of all terms from a₁ to aₙ. The nth term calculator provides this sum automatically when you enter the number of terms, saving you from manual arithmetic.

How does the geometric sequence calculator work?

The geometric sequence calculator uses the common ratio (r) to generate terms. Each term is the previous term multiplied by r. It also computes the sum using Sₙ = a₁ × (rⁿ - 1)/(r - 1) when r ≠ 1, or Sₙ = a₁ × n when r = 1. The tool displays the full sequence so you can verify the pattern visually.

Can I see arithmetic sequence examples?

Yes. A classic arithmetic sequence example is 3, 7, 11, 15, 19, where a₁ = 3 and d = 4. For a geometric sequence example, consider 2, 6, 18, 54, 162, where a₁ = 2 and r = 3. Use these values in the calculator to see how the nth term and sum are derived step by step.