What this symbol means?

Question

In the following quote, what does the notation $\{a_n\}$ mean?

Дана последовательность Фибоначчи $\{a_n\}$.

Translation: "You are given the Fibonacci sequence $\{a_n\}$."

Answer 1

The meaning of this notation here is the Fibonacci sequence . Which is exactly the sequence of terms 1,1,2,3,5,8.......

Answer 2

Short Answer:

The notation $\{a_n\}$ denotes a sequence. The first term of the sequence is $a_1$ (or $a_0$, if we index starting at zero), the second term is $a_2$, and so on. That is, $$ \{a_n\} = \{a_1, a_2, a_3, \dotsc\}. $$ Since we are told that $\{a_n\}$ is the Fibonacci^[1] sequence, this implies that $$ \{a_n\} = \{ 1, 1, 2, 3, 5, \dotsc\}, $$ that is, $a_1 = a_2 = 1$, and $a_{n+1} = a_n + a_{n-1}$ for $n \ge 2$.

Long Answer:

In the modern understanding of mathematics, a sequence is generally understood to be a function $$ a : \mathbb{N} \to X, $$ where $X$ is some set of interest. That is, a sequence is a function which takes a natural number as input, and outputs some element of a given set $X$. For example, consider a harmonic progression defined by $a : \mathbb{N} \to \mathbb{R}$ where $$ a(n) = \frac{1}{n}. $$ Because sequences appear so much in mathematics, there are a number of different notations for them. These various notations have developed for different reasons, and likely were regarded as distinct until the late 19^th or early 20^th Century, when modern mathematics and notation emerged. The following is not an accurate history of the development of notation, but does give a picture of how you might think about things.

A very natural notation is to simply list the terms (indeed, before the notion of a "function" was made formal, this was likely one of the best ways to discuss sequences): $$ 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dotsc. \tag{$\ast$}$$ The problem with this notation is that we might want to refer to a specific term in the sequence, and it becomes awkward to refer to "the $7$-th term of the sequence given at ($\ast$)" over and over again. We can, instead, name the sequence, e.g. $$ a := 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dotsc. $$ It is then possible to refer to "the $7$-th term of $a$" if we require. Or, even better, we just stick a subscript on $a$, and write $a_7$. This has the advantage of making the notation more concise, but also allows us to deal with much more complicated sequences where the pattern may not be readily apparent. The Fibonacci sequence provides a nice example: what is the next term of $$ F := 1, 1, 2, 3, 5, 8, 13, \dotsc? $$ The rule I am using to produce this sequence is not obvious. However, if I write "$F$ is a sequence defined by the formula $$ F_1 = 1, \qquad F_2 = 1, \qquad F_{n+1} = F_{n} + F_{n-1}," $$ then it is clear how the sequence is being produced.

At some point, it might be desirable to think of the sequence as a whole, rather than term-by-term. That is, we might want to think of the entire set of values attained by the sequence. Both curly braces and round braces are often used to do this. Generally, they can be interpreted to mean exactly the same thing, though I would argue that they have a slightly different flavor: $$ \{ a_n \} = \{ a_1, a_2, a_3, \dotsc \} $$ seems to denote the collection of all values attained by the sequence, whereas $$ (a_n) = (a_1, a_2, a_3, \dotsc ) $$ seems to denote an ordered list of values attained by the sequence. However, in the real world, these two notations are generally entirely equivalent.

One other bit of notation which may come up: it is also possible to define sequences as functions which have a countable (either finite or countably infinite) domain. When there is a chance for confusion, the domain is often included in a subscript. For example, $$ \left\{ \frac{1}{n} \right\}_{n\in\mathbb{N}} = \left\{ 1, \frac{1}{2}, \frac{1}{3}, \dotsc \right\} \qquad\text{or}\qquad \{n^2\}_{n\in\mathbb{Z}} = \{\dotsc,9,4,1,0,1,4,9,\dotsc\}. $$

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[1] For some reason, it amuses me that "Fibonacci" is transliterated into Russian as «Фибоначчи», retaining the double «чч». Orthography beats out pronunciation, I suppose?