Number Sequence Calculator

Number Sequence Calculator

Explore the most common types of number sequences with this easy-to-use calculator: arithmetic, geometric, and Fibonacci.

Arithmetic Sequence Calculator

Formula:
    an = a₁ + d × (n - 1)
Example:
    1, 3, 5, 7, 9, 11, 13, ...

Inputs:

• First term: 2

• Common difference (d): 5

• Find the 20th term

Geometric Sequence Calculator

Formula:
    an = a × rⁿ⁻¹
Example:
    1, 2, 4, 8, 16, 32, 64, ...

Inputs:

• First term: 2

• Common ratio (r): 5

• Find the 12th term

Fibonacci Sequence Calculator

Formula:
    a₀ = 0; a₁ = 1; an = an₋₁ + an₋₂
Example:
    0, 1, 1, 2, 3, 5, 8, 13, 21, ...

Input:

• Find the 25th term

Result:

• 25th term: 75,025

• Sum through 25th term: 196,417

What is a Number Sequence?

A number sequence is an ordered list of numbers that follow a specific pattern. Each value in the list is called a term, and the total count of values is the length of the sequence. The order is essential, and values may repeat depending on the pattern.

Sequences can be finite or infinite and are widely used in many areas of mathematics, including calculus and analysis, especially to study convergence. If a sequence approaches a specific value, it’s convergent; if not, it’s divergent.

Common Types of Sequences

1. Arithmetic Sequence

An arithmetic sequence increases or decreases by a constant difference between consecutive terms.

General Formula:
an = a₁ + d × (n - 1)

Where:

• an = nth term

• a₁ = first term

• d = common difference

• n = term number

Example:
1, 3, 5, 7, 9, 11... (d = 2)
To find the 5th term:
a₅ = 1 + 2 × (5 - 1) = 1 + 8 = 9

Sum of First n Terms:
Sum = (n × (a₁ + an)) / 2
Example sum of first 5 terms:
(5 × (1 + 9)) / 2 = 25

2. Geometric Sequence

In a geometric sequence, each term is multiplied by a fixed number (common ratio) to get the next.

General Formula:
an = a × rⁿ⁻¹

Where:

• an = nth term

• a = first term

• r = common ratio

• n = term number

Example:
1, 2, 4, 8, 16... (r = 2)
To find the 8th term:
a₈ = 1 × 2⁷ = 128

Sum of First n Terms (when r ≠ 1):
Sum = a × (1 - rⁿ) / (1 - r)
Example sum of first 3 terms:
1 + 2 + 4 = 7
1 × (1 - 2³) / (1 - 2) = -7 / -1 = 7

3. Fibonacci Sequence

Each term is the sum of the two previous terms, starting with 0 and 1 (or sometimes 1 and 1).

Formula:
a₀ = 0, a₁ = 1, an = an₋₁ + an₋₂

Example:
0, 1, 1, 2, 3, 5, 8, 13, 21...

Used in:

• Computer science (e.g., recursion, algorithms)

• Nature (e.g., leaf patterns, flower petals)

• Financial models

Related Calculator:
Sample Size Calculator, Rounding Calculator

 

External Resources:
Sequence Number Calculator on Calculator.net