Number Sequence Calculator

In mathematics, a sequence is an ordered list of objects. Accordingly, a number sequence is an ordered list of numbers that follow a particular pattern. The individual elements in a sequence are often referred to as terms, and the number of terms in a sequence is called its length, which can be infinite. In a number sequence, the order of the sequence is important, and depending on the sequence, it is possible for the same terms to appear multiple times. There are many different types of number sequences, three of the most common of which include arithmetic sequences, geometric sequences, and Fibonacci sequences.

Sequences have many applications in various mathematical disciplines due to their properties of convergence. A series is convergent if the sequence converges to some limit, while a sequence that does not converge is divergent. Sequences are used to study functions, spaces, and other mathematical structures. They are particularly useful as a basis for series (essentially descriptions of an operation of adding infinite quantities to a starting quantity), which are generally used in differential equations and the area of mathematics referred to as analysis. There are multiple ways to denote sequences, one of which involves simply listing the sequence in cases where the pattern of the sequence is easily discernible. In cases that have more complex patterns, indexing is usually the preferred notation. Indexing involves writing a general formula that allows the determination of the n^th term of a sequence as a function of n.

Arithmetic Sequence

An arithmetic sequence is a number sequence in which the difference between each successive term remains constant. This difference can either be positive or negative, and depending on the sign will result in terms of the arithmetic sequence tending towards positive or negative infinity. The general form of an arithmetic sequence can be written as:

a_n = a₁ + f × (n-1)   or more generally   a_n = a_m + f × (n-m)

i.e. a₁, a₁ + f, a₁ + 2f, ... where a_n refers to the n^th term in the sequence, a₁ is the first term, and f is the common difference.

EX: 1, 3, 5, 7, 9, 11, 13, ... — the common difference f is 2. Using the equation above to calculate the 5^th term: a₅ = 1 + 2 × (5-1) = 1 + 8 = 9, which matches the listed sequence as expected. It is also commonly desirable, and simple, to compute the sum of an arithmetic sequence using the following formula in combination with the previous formula to find a_n:

n × (a₁ + a_n)2

Using the same number sequence in the previous example, find the sum of the arithmetic sequence through the 5^th term: 1 + 3 + 5 + 7 + 9 = 25, and (5 × (1 + 9))/2 = 50/2 = 25.

Geometric Sequence

A geometric sequence is a number sequence in which each successive number after the first number is the multiplication of the previous number with a fixed, non-zero number (common ratio). The general form of a geometric sequence can be written as:

a_n = a × r^n-1

i.e. a, ar, ar², ar³, ... where a_n refers to the n^th term in the sequence, a is the scale factor, and r is the common ratio.

EX: 1, 2, 4, 8, 16, 32, 64, 128, ... — the common ratio r is 2, and the scale factor a is 1. Using the equation above, the 8^th term: a₈ = 1 × 2⁷ = 128, which confirms the value in the listed sequence. The equation for calculating the sum of a geometric sequence:

a × (1 − rⁿ)1 − r

Using the same geometric sequence above, find the sum through the 3^rd term: 1 + 2 + 4 = 7, and 1 × (1−2³)1 − 2 = −7−1 = 7.

Fibonacci Sequence

A Fibonacci sequence is a sequence in which every number following the first two is the sum of the two preceding numbers. The first two numbers in a Fibonacci sequence are defined as either 1 and 1, or 0 and 1 depending on the chosen starting point. Fibonacci numbers occur often, as well as unexpectedly within mathematics, and are the subject of many studies. They have applications within computer algorithms (such as Euclid's algorithm to compute the greatest common factor), economics, and biological settings including the branching in trees, the flowering of an artichoke, as well as many others. Mathematically, the Fibonacci sequence is written as:

a_n = a_n-1 + a_n-2   where a₀ = 0; a₁ = 1

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

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