A variable for the first 8 integers? I wish to use algebra to (is the term truncate?) the set of positive integers to the first 8 and call it for example 'n'.
In order to define $r_n = 2n$ or similar.
This means:
$$r_0 = 0$$
$$r_1 = 2$$
$$\ldots$$
$$r_7 = 14$$
However there would not be an $r_8$.
edit: Changed "undefined" to "would not be", sorry about this.
 A: You could simply write

"Let $r_n = 2n$ for all integers $n$ from $0$ to $7$."

or perhaps

"Let $r_n = 2n$ for $n \in \{0,1,2,\dotsc,7\}$."

or, using the compact notation suggested by Ilya,

"Let $r_n = 2n$ for $n \in \overline{0,7}$."

or even, if it's clear from context that $n$ is an integer,

"Let $r_n = 2n$ for $0 \le n \le 7$."

However, if going with Ilya's notation, be aware that many English-speaking readers may not be familiar with it, so you should define it yourself, e.g. "Let $\overline{n,m} = \{n,\,\dotsc,\,m\}$ denote the set of integers between $n$ and $m$ inclusive."
(Ps. What's wrong with MathJax's rendering of \overline, anyway?  The lines in my examples above seem to curve upwards at the ends like this, while the first — but not the second — one in Ilya's comment curves the other way.)
A: $r_n = 2n$ if $0 ≤ n ≤ 7$
Thanks Ilmari Karonen for the correction.
A: You've tagged this abstract-algebra and group-theory but it's not entirely clear what you mean.
However, by these tags, perhaps you are referring to $\left(\Bbb Z_8, +\right)$?
In such a case, you have $r_1+r_7 = 1+_8 7 = 8 \mod 8 = 0$. So there is no $r_8$ per se; however, the re-definition of the symbols $r_0, r_1, \ldots$ is superfluous.
