Difference between letters used as indices of arbitrary elements in a matrix/vector and letters used as indices of the last element in a matrix/vector

Question

Let's say I want to define a vector $A$.

When I intend to use a particular letter (eg. letter $j$) to index an arbitrary element of vector $A$, can I use the same letter to index the last element of vector $A$ when defining the vector? Like this:

$$A = (a_{1}, a_{2}, ... ,a_{j})$$ where $j$-th element means such and such...

Or do I have to use another letter to index the last element of vector $A$? Like this:

$$A = (a_{1}, a_{2}, ... ,a_{n})$$ where $j$-th element means such and such and where $j \in \text{{$1; 2; ... ;n$}}$

Or could I do this? $$A = (a_{1}, a_{2}, ... ,a_{N^{(j)}})$$ where $j$-th element means such and such and where $j \in \text{{$1; 2; ... ;N^{(j)}$}}$

If I indeed have to use different letters to index an arbitrary element and the last element of vector $A$, can I use for example $J$ to index the last element when using $j$ to index an arbitrary element of vector $A$? Like this: $$A = (a_{1}, a_{2}, ... ,a_{J})$$ where $j$-th element means such and such and where $j \in \text{{$1; 2; ... ;J$}}$

Answer 1

You are overthinking this and will confuse your reader. There is no need for the superscript $(j)$ in $$ A = (a_{1}, a_{2}, ... ,a_{N^{(j)}}). $$ Write simply $$ A = (a_{1}, a_{2}, ... ,a_{N}). $$ Your vector has $N$ components, independent of any choice you make for how to label an arbitrary component. Then you write $$ a_j \ \ (j \in \{1, 2, \ldots, N\}) . $$ The last element in the vector is simply $a_N$.