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I have a more general question to ask:
about using the $\cdots$ symbol.

As far as I have heard three dots will represent that it continues up to infinity.

And I have found (mostly in computer programming) that two dots will be used in a case to indicate the terms between two specific points.

So my question is what is suitable to write in a case where finite number of terms are used.

For example Consider the set with $n$ number of elements. Let them be
$$x_1,x_2, \cdot \cdot x_n \;\text{ two dots. }$$ Or is it $$x_1,x_2, \cdots x_n\;\;?$$

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    Three dots is the convention for both cases. Also note that including a comma following the dots is conventional as well: $x_1, x_2, \dots, x_n$ – 2017-01-19
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    So isn't the three dots represent that it continues up to an infinite value? – 2017-01-19
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    To imply something continued to infinity you would write $x_1 ,x_2 ,x_3 ,\ldots$ but for something finite you use $x_1 ,x_2 ,x_3 ,\ldots ,x_n$ – 2017-01-19
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    @CharithJeewantha No, three dots indicates a finite set of intermediate values as I indicated. Your idea isn't bad, and in fact you could define your own notation that way and it would make perfect logical sense. It just happens to be conventional to write three dots in the middle even when it only stands for finitely many values. – 2017-01-19
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    Computer languages use notation that is related to mathematical notation but different. When programming, we need something that is easy to read as a linear string of symbols, printed in monospaced font from a very limited set of characters. In math, we can arrange things in two dimensions, we can draw from a much richer set of glyphs, and we want something that is pleasant to read when it is typeset in a professional manner. – 2017-01-19
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    "..." is not a symbol at all and is not any formal mathematical notation at all that I know of. It is merely a convention of clarity to explain there is a pattern of indefinite length, infinite or finite. That computer programmers use two dots is merely because the created a code that can *mechanically* interpret and calculate such a meaning. – 2017-01-19

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You'll need to use three dots:

$\{1, 2, 3, \ldots \}$

$\{1, 2, \ldots, 10\}$

You can also use three dots to indicate an infinite sum, or to abbreviate the sum of many numbers, provided you've made the pattern clear.

$\sum_{k = 1}^{k=119} k = 1 + 2+ 3+ \cdots + 119\\$

$\sum_{k=1}^\infty k =1 + 2 + 3 + \cdots$

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    Notice also that the dots are at the same height as the commas in the comma-separated lists, but they are (usually) put at the same height as the middle of the $+$ signs in sums. – 2017-01-19
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    Are you pointing this out to the OP? Because that is precisely what I did. And if so, thanks for drawing attention to that, since I never explicitly pointed that out. – 2017-01-19
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    Note also, Charith, that the three dots are so conventional that MathJax responds to `$\ldots$` as $\ldots$, meaning "low dots" used when listing a comma separated list. And it responds to `$\cdots$` as $\cdots$, meaning "center dots", used when indicating a repetition of an operator. I used this in the infinite sum and the finite sum above. But $\cdots$ can also be used with other operations: $1\times 2\times 3 \times \cdots \times n = n!$, in the finite case, and $ 1\times 2\times 3 \times \cdots$. – 2017-01-19
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    Yes, that was to the OP to "notice also" how the dots in your answer were formatted, which is how people usually do it. – 2017-01-20