A sequence in mathematics is an enumerated collection of items in which repetitions are permitted and order is important. It, like a set, has members (also called elements, or terms). Sequences can be finite, as in the examples above, or endless, as in the series of **all even positive numbers** (2, 4, 6,).

Every finite sequence has a last element, and since repetition is allowed, this last element may itself be repeated. Thus the sequence {1, 2, 3, 4} contains four elements, but the sequence {1, 1, 1} contains only three: two identical elements followed by one different one. This difference is what makes sequences interesting. There are many ways to define equality between elements of a sequence, but for our purposes it will be enough to say that two elements are equal if and only if they are the same position in their respective sequences.

It follows that every sequence must contain **at least one element** which is not equal to any other element. Such an element is called a singular term or a singular point of the sequence. If there are no singular points in a sequence, then that sequence is said to be infinite. For example, the sequence of **natural numbers** includes 0, which is not equal to **any other number** in the sequence. Therefore, the sequence is infinite.

Some sequences have more than one singular point.

A sequence is a list that has been sorted. It is a function with the natural numbers 1, 2, 3, and 4 as **its domain**. Each number in a series is referred to as a term, element, or member. The digits of p, for example, constitute a sequence but do not follow a pattern. On the other hand, the numbers 1, 4, 9, 16, 25, 36, 49, 64 are a pattern but not a sequence. Patterns can be very useful for guessing what number will be next in a series.

All sequences have a pattern. This pattern is called the repeating decimal. It is based on the digits from 0 to 9 and looks like this:... n where n goes over 10. For example, the repeating decimal 0.123456789101112131415161718192021 contains 8 ones, 5 twos, and so on.

The repeating decimal shows how any sequence can be broken down into **its individual terms**, which are then added together. For example, the sum of **the first five terms** of 0.123456789101112131415161718192021 is 552.23 because 0+1+2+3+4=11 and so on.

The reason why all sequences have a repeating decimal is because every number can be expressed as a simple fraction with **only two digits**: 0.000.. .000.. .0001..

An infinite sequence is a set of discrete objects, generally numbers, that can be paired off one-to-one with the set of positive integers 1, 2, 3. N = (0, 1, 2, 3,) and S = (1, 1/2, 1/4, 1/8, 1/2 n,) are two examples of infinite sequences. In mathematics, an example or instance of a thing is a representation of that thing, such as a diagram or list. So, an example of an infinite sequence is something like this: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194320, 8388608, 16717952, 33478688, 66914336.

Infinite sequences appear in **many places** in mathematics and science. They are important in **number theory** because they provide examples for many problems about sets of natural numbers. For instance, it can be shown that if an infinite sequence contains a copy of every natural number greater than or equal to 2 then it must contain either all natural numbers or no natural numbers at all. Another interesting property of infinite sequences is that there cannot exist any first element or first few elements that are repeated infinitely often; otherwise, we could divide into cases where some elements are repeated finitely many times and others are repeated infinitely many times which would be contradictory.

Number sequences are groups of numbers that adhere to a pattern or guideline. An arithmetic sequence is one in which the rule is to add or remove a number each time. A linear series is one that grows or lowers by the same amount each time....

There is a first term, a second term, and so on in an infinite series. The nth phrase in a series is commonly represented as a (n). For example, the first term of a series is a (1), while the 23rd term is a (23). Subscripts are the numerals in parenthesis next to the a. In this case, it means that there are twenty-three terms in the series.

An infinite sequence has no final term. It can go on forever because there will always be another item in the sequence for it to be written down. Writing an infinite sequence is similar to writing a circle because they both start at the same place and finish back where they began. A circle ends when it returns to its starting point while an infinite sequence never ends.

There are two ways to write down **an infinite sequence**: indexing from the beginning or counting up from one. If we wanted to list all the integers between 1 and 10 we could write down each one like this: 1, 2, 3, 4, 5, 6, 7, 8, 9, or we could write down **every single number** from 11 right after 10. Indexing starts with 1 instead of 0; it goes until it reaches the last term of the sequence. Counting starts with 1 and keeps going until you get to the last term too. Both methods work well for **infinite sequences** because they give you the same results without having to think about how many terms are in the sequence.

If a series contains a finite number of terms, it is finite; otherwise, it is endless. The initial number in the sequence is four, and the last term is sixty-four. The series is finite since it has a last term. The number series is endless: 4, 8, 12, 16, 20, 24,...

A finite sequence has a fixed total number of terms, a beginning number, a difference or factor, and a fixed total number of terms. Infinite sequences have **an infinite number** of phrases that can increase to infinity, decline to zero, or approach a fixed value. The outcome of the related series can either be infinite, zero, or fixed. An example of an infinite sequence is the list of integers greater than or equal to two: 2, 3, 4, 5..