The first number of terms of a sequence The task is to list the first 10 terms of the sequence whose $n$th term is $n!-2^n$. Easy enough, except to me there's some ambiguity regarding the initial term. I happened to start at $n=0$ because the general term is defined for $n=0$. Thus my initial term is 0. But the book's initial term is $-1$, meaning it started at $n=1$. How do you know whether to start at $n=1$ or $n=0$? Is the fact that they asked for the first 10 terms indicate that you start $n$ at 1? Like if they said, "list the zeroth 10 terms," then you start $n$ at $0$? Or is it just "convention" to start at $n=1$ unless otherwise noted?
Edit: Webster's Dictionary defines "first" as "coming before all others in time, order, or importance." I don't know if the English meaning translates to math, but if it did, "first" is synonymous with "initial." So "first 10 terms" could be read, "initial 10 terms." In which case, the first term of this sequence could be $0$ or $-1$, depending on how the sequence is indexed. In that case there would be ambiguity, unless there was an understood convention regarding indexing.
 A: If the $\color{red}n$th term is $\color{red}n!-2^{\color{red}n}$, then the $\color{red}1$st term is $\color{red}1!-2^{\color{red}1}$.
The (one) first term is the first term; the two first terms are the first and the second term, ...
This may not make set theorista happy (or at least make them disagree), but we need to translate a linguistic expression here.
A: It's just convention to start at $n=1$. Unfortunately, they should be more explicit. For example, a sequence is usually defined as
$$(a_k)_{k \in \mathbb{N}}$$
where $\mathbb{N}$ denotes the set of natural numbers. Typically that does not include $0$ and when one wants to express the set of non-negative integers (i.e. including zero) one denotes it $\mathbb{N}_0$. Though this notation is not universal and most books should be more explicit about it.
A: It does not really matter, so long as you are consistent about what the first, second, ... $n^\text{th}$, ... tenth, term means.
If you use start-one-indexing, then you wish to find the series $\{a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_{10}\}$ for which you know the $n^\text{th}$ term ($a_{n}$) is equal to $n!-2^n$.
If you use start-zero-indexing, then you wish to find the series $\{a_0, a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9\}$ for which you know the $n^\text{th}$ term ($a_{n-1}$) is equal to $n!-2^n$.
Proceed.
