In mathematics, a sequence is a list of numbers written in a specific order. Sequences can be either finite, meaning they contain a finite number of terms, or infinite, meaning they continue indefinitely. There are many ways to define a sequence.
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Using words — In this case, each term of the sequence shares a particular characteristic or quality that can be described using regular language. Some examples of this type of sequence include: the prime numbers {2, 3, 5, 7, 11...}, the eban numbers (the natural numbers whose English names do not contain the letter E) {2, 4, 6, 30, 32...}, and the number of letters in the English names of the natural numbers {3, 3, 5, 4, 4, 3...}.

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Using a closed (explicit) formula — In this case, each term of the sequence is the value of a specific function evaluated at n, that term's index or position in the sequence, and so we can write ${a}_{n}equals;f\left(n\right)$, where n belongs to some subset of the natural numbers. Some examples of this type of sequence include:

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The arithmetic sequence ${a}_{n}equals;{a}_{1}plus;\left(n1\right)\cdot d$ where ${{a}_{1}}_{}$ is the first term and d is the common difference between each of the terms

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The geometric sequence ${a}_{n}equals;a\cdot {r}^{n1}$ where a is the initial value and r is the common ratio

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The Harmonic sequence ${a}_{n}equals;\frac{1}{n}$

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Using recursion — In this case, each consecutive term of the sequence is constructed using the preceding terms, with enough initial conditions provided to apply this rule to specify the first recursive term. Some examples of this type of sequence include:

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The arithmetic sequence ${a}_{n}equals;{a}_{n1}plus;d$

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The geometric sequence ${a}_{n}equals;{a}_{n1}\cdot r$

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The Fibonacci sequence ${a}_{n}equals;{a}_{n1}plus;{a}_{n2}$ with ${a}_{0}equals;0$ and ${a}_{1}equals;1$

A sequence is said to converge if there is a limit L such that the terms (the ${a}_{n}$'s) become arbitrarily close to L as the number of terms, N, grows very large.
So, a convergent sequence has a numeric limit as n approaches infinity: $\underset{n\to \infty}{lim}{a}_{n}equals;L$. If a sequence does not converge, it is said to diverge.
If the ${a}_{n}$'s get arbitrarily large as n approaches infinity, we write $\underset{n\to \infty}{lim}{a}_{n}equals;\infty$, and we can say that the sequence {${a}_{n}$} diverges or converges to infinity.
If the ${a}_{n}$'s get arbitrarily large and negative as n approaches infinity, we write $\underset{n\to \infty}{lim}{a}_{n}equals;\infty$, and we can say that the sequence {${a}_{n}$} diverges or converges to negative infinity.
