nth term calculator

What Is the n-th Term?

The n-th term is the value at position n in a sequence. Instead of listing every value one by one, you can use a direct formula to jump to any position. This is especially useful in algebra, exam problems, and real-life growth/decline models.

Two Common Sequence Types

1) Arithmetic Sequence

In an arithmetic sequence, each term changes by the same amount (called the common difference).

• Example sequence: 5, 8, 11, 14, 17, ...
• Common difference: d = 3
• Formula: an = a1 + (n - 1)d

2) Geometric Sequence

In a geometric sequence, each term is multiplied by the same factor (called the common ratio).

• Example sequence: 2, 6, 18, 54, ...
• Common ratio: r = 3
• Formula: an = a1 × r(n - 1)

How to Use This n-th Term Calculator

1. Select the sequence type: arithmetic or geometric.
2. Enter the first term a₁.
3. Enter the common difference d or ratio r.
4. Enter n, the position you want to find.
5. Click Calculate to get the exact n-th term and a preview of early terms.

Worked Examples

Example A: Arithmetic

Suppose a₁ = 7, d = 5, and n = 12.

a12 = 7 + (12 - 1) × 5 = 7 + 55 = 62

The 12th term is 62.

Example B: Geometric

Suppose a₁ = 3, r = 2, and n = 8.

a8 = 3 × 27 = 3 × 128 = 384

The 8th term is 384.

Common Mistakes to Avoid

• Using n instead of n - 1 in the formula.
• Mixing up arithmetic and geometric patterns.
• Forgetting that n must be a positive integer (1, 2, 3, ...).
• Entering a ratio when the pattern is additive, or a difference when the pattern is multiplicative.

Why This Matters

n-th term formulas appear in finance, computer science, physics, and statistics. They help model savings schedules, exponential growth, depreciation, repeated processes, and pattern prediction.