What is a Number Sequence?
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 a "term", 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 describe 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.
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 dependent on the sign will result in terms of the arithmetic sequence tending towards positive or negative infinity.
Variables:
• an = the nth term in the sequence
• a1 = the first term
• f = the common difference
• n = the term position
It is also commonly desirable, and simple, to compute the sum of an arithmetic sequence. The formula for the sum of an arithmetic sequence up to the nth term is:
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 known as the common ratio.
Variables:
• an = the nth term in the sequence
• a = the scale factor (first term)
• r = the common ratio
The equation for calculating the sum of a geometric sequence is as follows:
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.
Base Cases: a0 = 0, a1 = 1
Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
Frequently Asked Questions
A number sequence is an ordered list of numbers that follow a particular pattern or mathematical rule. Each individual number in the sequence is called a "term".
An arithmetic sequence is one where the difference between consecutive terms is constant. For example, in the sequence 2, 5, 8, 11..., the common difference is 3.
A geometric sequence is one where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in 2, 6, 18, 54..., the common ratio is 3.
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. The sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.
A sequence is simply a list of numbers arranged in an order (e.g., 1, 2, 3, 4), while a series is the sum of those numbers (e.g., 1 + 2 + 3 + 4). Our calculator provides both the nth term (sequence item) and the sum (series) up to that term.