# Number Sequence Calculator

### Arithmetic sequence calculator

Formula: a_{n} = a_{1} + (n-1)d

example: 1, 3, 5, 7, 9, 11, 13, ...

### Geometric Sequence Calculator

Formula: a_{n} = a_{1} (r)^{n-1}

example: 1, 2, 4, 8, 16, 32, 64, 128, ...

### Fibonacci Sequence Calculator

Formula: a_{n} = a_{n-1} + a_{n-2}

example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

### Number sequence

In mathematics, an ordered list ruled by a certain pattern is called a number sequence. For this type of sequence, the order is extremely important, as the same terms can appear more than once.

The individual components on the list are called terms, whereas the number of terms constitutes the sequence length. Depending on it, sequences can be either finite or indefinite, meaning they can go on without terminating.

While there are numerous types of number sequences, the three most popular are:

- Arithmetic sequences
- Geometric sequences
- Fibonacci sequences.

Number sequences have multiple applications in math since they converge to a certain limit. Mathematicians often use sequences to study different structures – spaces, functions, etc. Mathematical analysis and differential equations are other fields where sequences are commonly applied. They are utilized to constitute a basis for series, which then apply to various equations.

As for denoting number sequences, there are a few options. If the pattern is easily detectable, you can just list the terms of a sequence. And in more complex cases, where the sequence pattern is obscure, mathematicians use indexing.

### Arithmetic sequence

An arithmetic sequence is a number sequence where every next term is created by adding a constant number to the previous one. Thus, the difference between each subsequent number remains constant as well. Furthermore, the common difference can be either positive or negative. This, in its turn, determines whether the sequence will lead toward positive or negative infinity.

In general, arithmetic sequences can be visualized in this formula:

a_{n} = a_{2} + f × (n-1) or a_{n} = am + f × (n-m),
where an is the nth number in the whole sequence, a1 is the first term, and f is the difference between them.

So, a typical arithmetic sequence can look like this: 1, 3, 5, 7, 9, 11, 13, ... Here, the common difference, or f, is 2. We can use the equation above to determine the fifth term in the sequence:

a_{5} = a_{1} + f × (n-1);

a_{5} = 1 + 2 × (5-1);

a_{5} = 1 + 8 = 9;

### Geometric sequence

A geometric sequence is a sequence of numbers, where each next term after the first one is created by multiplying the following number by a constant. This constant or a fixed number is also called a common ratio and cannot be a zero.

Typically, geometric sequences can be denoted in the following equation:

a_{n} = a × r^{n-1}, where a_{n} is the n^{th} term in the sequence, a refers to the scale factor, and r is the common ratio.

So, it goes like this: a, ar, ar^{2}, ar^{3}, ... For example. 1, 2, 4, 8, 16, 32, 64, 128, ... - this is a geometric sequence, where the common ratio is 2.

### Fibonacci sequence

A Fibonacci sequence is a sequence of numbers in which each next number is a sum of two previous numbers. Depending on their starting point, Fibonacci sequences begin with either 0 and 1 or 1 and 1.

Fibonacci numbers are used commonly and unexpectedly within the mathematical realm. They are often applied in economics, biology, and computer studies. Some of the most popular uses of Fibonacci numbers occur in Euclid's algorithm and for branching trees or artichoke flowering.

The formula used to write a Fibonacci sequence mathematically is:

a_{n} = a_{n-1} + a_{n-2}, where a_{n} denotes the n^{th} term in the sequence. For instance, 0, 1, 1, 2, 3, 5, 8, 13, 21, ...