Free Sequence Calculator

What Is a Number Sequence?

A number sequence is an ordered collection of numbers where the position matters, values may repeat, and each term can be seen as the output of a function indexed by its place. Terms are usually denoted with the letter $a$ followed by a subscript indicating the position, e.g., $a_0$ is the first term, $a_1$ the second, and $a_n$ the term at index $n$. This number sequence calculator online serves as both an arithmetic sequence solver and a geometric sequence calculator, and it includes a Fibonacci sequence generator as well as a prime number sequence finder — all in one tool.

Arithmetic Sequences

In an arithmetic sequence, consecutive terms differ by the same constant quantity $d$. Formally, $a_{n} - a_{n-1} = d$ for all $n$. Starting from a first term $a_0$, any term can be obtained by adding $d$ repeatedly. The direct formula for the nth term (starting at $n = 0$) is:

$$a_n = a_0 + n \times d$$

If you know the value at some other index and the common difference, you can compute $a_0$ as well. For example, with $d = 5$ and $a_0 = 0$, the first ten terms are 0, 5, 10, 15, 20, 25, 30, 35, 40, 45. On a graph, these points lie on a straight line whose slope equals $d$. When using the arithmetic sequence solver part of this calculator, you supply the starting term and the difference, and it outputs the sequence for any requested index.

Geometric Sequences

A geometric sequence is defined by a constant ratio $r$ between adjacent terms: $a_{n} / a_{n-1} = r$. To move forward, multiply the previous term by $r$; to move backward, divide. The nth term (index starting at 0) is:

$$a_n = a_0 \times r^{n}$$

If $r > 1$, the sequence grows extremely quickly, resembling exponential growth; if $0 < r < 1$, it decays toward zero. For instance, with $a_0 = 1$ and $r = 2$, the powers of two appear: 1, 2, 4, 8, 16, 32, … . The geometric sequence calculator mode accepts the initial term and the ratio and returns the requested terms.

Figurate Numbers: Squares, Cubes, and Triangles

Figurate numbers can be arranged into regular geometric shapes. The most famous are square numbers, where the nth square is simply $n^{2}$. Cube numbers are $n^{3}$. Triangular numbers are given by:

$$T_n = \frac{n(n+1)}{2}$$

The table below compares the first ten values of these three figurate sequences (starting from $n = 0$):

n	Square ($n^2$)	Cube ($n^3$)	Triangular ($\frac{n(n+1)}{2}$)
0	0	0	0
1	1	1	1
2	4	8	3
3	9	27	6
4	16	64	10
5	25	125	15
6	36	216	21
7	49	343	28
8	64	512	36
9	81	729	45

Notice the rapid increase in the cube column compared to the square column. Triangular numbers grow at a moderate pace. A neat relationship exists: the sum of two consecutive triangular numbers equals a square number, which can be verified from the table (e.g., $T_4 + T_5 = 10 + 15 = 25 = 5^2$).

Prime Numbers

Prime numbers are integers greater than 1 that have no positive divisors except 1 and themselves. The prime number sequence begins 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, … . Neither 1 nor 0 is considered prime (though 1 was historically debated). Finding large primes usually requires advanced algorithms, but tabulated lists are widely available. The prime number sequence finder integrated into this calculator lets you generate primes up to a chosen limit.

Fibonacci Numbers

The Fibonacci sequence follows a recurrence relation where each term equals the sum of the two preceding terms:

$$F_n = F_{n-1} + F_{n-2}$$

By convention, $F_0 = 0$ and $F_1 = 1$. This produces the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … . The growth is approximately exponential after the first few terms. The Fibonacci sequence generator in this calculator requires the first two terms (defaulting to 0 and 1) and can produce any desired number of subsequent terms.

Other Important Sequences

Powers of Two
This is essentially a geometric sequence with $a_0 = 1$ and $r = 2$: 1, 2, 4, 8, 16, 32, 64, … . Powers of two are fundamental in computing and binary systems.

Factorial Numbers
The factorial of a non‑negative integer $n$, denoted $n!$, is the product of all positive integers from 1 to $n$ (with $0! = 1$ by definition). The sequence of factorials begins: 1, 2, 6, 24, 120, 720, 5040, … . Factorials grow extremely fast and are vital in combinatorics and probability.

Star Numbers
Star numbers form another class of figurate numbers; they represent the number of dots that can be arranged in a six‑pointed star. The nth star number (with $n$ starting at 1) is:

$$S_n = 6n(n-1) + 1$$

The first few star numbers are 1, 13, 37, 73, … . For example, the arrangement for $n = 3$ contains 37 points.

Using the Sequence Calculator

To get the most out of this number sequence calculator:

1. Choose the sequence type from the dropdown — arithmetic, geometric, Fibonacci, prime, power of two, factorial, or star.
2. Set the starting index (the position from which terms should be displayed).
3. Provide the required parameters:
   • Arithmetic: the first term $a_0$ and the common difference $d$.
   • Geometric: the first term $a_0$ and the common ratio $r$.
   • Fibonacci: the first two terms (default $F_0 = 0$, $F_1 = 1$).
   • Prime: enter the upper limit.
   • Others: may require an index or start value.
4. Review the results: the calculator outputs the first five terms starting from the specified index, along with the formula used.

The calculator also works in reverse: if you supply a term that appears in the sequence, it may infer the missing parameters. Remember that a sequence is distinct from a series, which is the sum of its terms. If you need summed values, consider using a series calculator.