# Number Sequence Calculator

Find the nth term and sum of arithmetic or geometric sequences. Enter the first term, common difference or ratio, and n. Free sequence calculator.

## What this calculates

Calculate the nth term and sum of arithmetic or geometric sequences. Enter the first term, the common difference (or ratio), and which term you want to find.

## Inputs

- **Sequence Type** — options: Arithmetic (constant difference), Geometric (constant ratio) — Select the type of sequence.
- **First Term (a₁)** — The first term of the sequence.
- **Common Difference or Ratio** — Common difference (arithmetic) or common ratio (geometric).
- **Find the nth Term (n)** — min 1 — Which term number to find.

## Outputs

- **nth Term (aₙ)** — The value of the nth term.
- **Sum of First n Terms (Sₙ)** — The sum of the first n terms.
- **Term Formula** — formatted as text — The general formula for the nth term.
- **First 8 Terms** — formatted as text — The first several terms of the sequence.
- **Sum Formula** — formatted as text — The formula used to compute the sum.

## Details

A number sequence is an ordered list of numbers that follows a specific pattern. The two most common types are arithmetic and geometric sequences.

**Arithmetic Sequences:**

Each term differs from the previous by a constant amount called the common difference (d).

- nth term: a_n = a₁ + (n-1)d
- Sum of first n terms: S_n = n/2 x (a₁ + a_n) = n/2 x (2a₁ + (n-1)d)

Example: 3, 7, 11, 15, 19... has a₁ = 3, d = 4. The 10th term is 3 + 9(4) = 39. The sum of the first 10 terms is 10/2 x (3 + 39) = 210.

**Geometric Sequences:**

Each term is multiplied by a constant ratio (r) to get the next term.

- nth term: a_n = a₁ x r^(n-1)
- Sum of first n terms: S_n = a₁(1 - r^n)/(1 - r), when r is not 1

Example: 2, 6, 18, 54... has a₁ = 2, r = 3. The 5th term is 2 x 3⁴ = 162. The sum of the first 5 terms is 2(1 - 243)/(1 - 3) = 242.

**Identifying the Type:**

If the differences between consecutive terms are constant, it is arithmetic. If the ratios are constant, it is geometric. If neither, the sequence follows a different pattern.

## Frequently Asked Questions

**Q: What is an arithmetic sequence?**

A: An arithmetic sequence has a constant difference between consecutive terms. For example, 5, 8, 11, 14, 17 is arithmetic with common difference 3. Every term equals the previous term plus 3. The nth term formula is a_n = a₁ + (n-1)d.

**Q: What is a geometric sequence?**

A: A geometric sequence has a constant ratio between consecutive terms. For example, 3, 6, 12, 24, 48 is geometric with ratio 2. Every term is twice the previous one. The nth term formula is a_n = a₁ x r^(n-1). Geometric sequences can grow very fast (or shrink if |r| < 1).

**Q: How do I find the sum of the first n terms?**

A: For arithmetic: S = n/2 x (first term + last term). For geometric: S = a₁(1 - r^n)/(1 - r). For the arithmetic sequence 2, 5, 8, 11, 14 (5 terms): S = 5/2 x (2 + 14) = 40. For the geometric sequence 1, 2, 4, 8, 16 (5 terms): S = 1(1 - 32)/(1 - 2) = 31.

**Q: Can a geometric sequence have a negative ratio?**

A: Yes. A negative ratio means the terms alternate in sign. For example, 1, -3, 9, -27, 81 has ratio -3. The signs flip between positive and negative, but the absolute values still follow the geometric pattern. All the same formulas apply.

**Q: What if the common ratio equals 1?**

A: When r = 1, every term in the geometric sequence is the same as the first term. The sequence is constant: a₁, a₁, a₁, ... The sum of n terms is simply n times a₁. The usual geometric sum formula does not apply when r = 1 (it would involve division by zero).

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Source: https://vastcalc.com/calculators/math/number-sequence
Category: Math
Last updated: 2026-04-08