# Sequence with a fixed last element Notation

I was trying to write a sequence of two different elements (that always appear in order) with a fixed last element, for an example: $A_1, B_1, A_2, B_2, A_3, B_3, A_4$. I'm not sure which would be the best way to explain and represent this situation.

The best that I reached was: $\Gamma = \langle (A_{i},B_{i})_{i=1}^{n},A_{n+1} \rangle$. But I'm not sure if this notation is correct because it is actually a sequence of ordered pairs with a last element (that is not an ordered pair).

So, I'm asking you, is it a valid way to represent what I want? If not, which would be the best way to write that?

I think you can safely go with something simple: $$A_1, B_1, A_2, B_2, \dotsc, A_n, B_n, A_{n+1}.$$

• Hum, it makes sense. Why simplify when we can complicate :D. My only doubt is, with this notation, could I have a sequence with only 3 elements? A1,B1,A2? Or since I have the n>1 I would need at least 5 elements? – copenhagen Dec 14 '13 at 6:26
• @Andre, you could clarify with "for $n\ge 1$" or something. Mathematical notation doesn't necessarily stand on its own. – dfeuer Dec 14 '13 at 6:34
• Sure. Let's say, if I define the sequence as $A_1,B_1,...,A_n,B_n,A_{n+1}$, $n$ will need to be greater than one, naturally. What I'm asking is if with this definition the sequence $A_1,B_1,A_2$ would be valid (because in this case I would not have any possible element as $A_n$ or $B_n$). – copenhagen Dec 14 '13 at 6:39
• @Andre, I understand your concern. On the other hand, using too few terms can be confusing in some cases. If you explicitly set the lower bound for $n$ that you want, I think whoever reads it will understand that if $n=1$ then the $A_2$ and $B_2$ terms will be dropped. In fact, if you were to write "for $n\ge 0$", then they would understand that if $n=0$ then the sequence will have no terms. – dfeuer Dec 14 '13 at 6:50
• First, thanks for your help. My concern actually is with the $A_n$ and $B_n$ terms that I need to keep in the definition. If I put "for $n \geq 0$" then a sequence $A_1,B_2,A_0,B_0,A_1$ would be valid. But I don't see a way to create a definition that could accept the $A_1,B_1,A_2$ sequence (since I can have a sequence with only 3 elements) and also be generic for other cases. Maybe $\langle A_i,B_i,A_{n+1} \rangle_{i=1}^{n}$ ? – copenhagen Dec 14 '13 at 7:00