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I would like clarification on the following definition of finite arithmetic progression:

According to Wikipedia, "A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series." https://en.wikipedia.org/wiki/Arithmetic_progression

Is there a minimum as to how many terms ($a+nd$) must belong to a progression before it's "finite"? It would be at least two, right? For example: $a+nd$, $a+$($n+1$)$d$ = 3, 23 (where $a$ = 3, $d$ = 20, $n \geq 0$).

M. B. Jones
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    You could consider a single number a part of an arithmetic progression. In fact, a single number is part of infinitely many arithmetic progressions. In Mathematics many times definitions are taken to capture some notions and as a result, some extreme cases are included, even if they feel a little weird. – jjagmath Nov 26 '23 at 03:48
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    Being an arithmetic progression is only interesting for sequences with at least three terms. If there is just one term, then the common difference does not make sense. For sequences with two terms, we could say that they are arithmetic sequences, but only in a trivial sense. – Geoffrey Trang Nov 26 '23 at 03:50
  • @GeoffreyTrang A sequence a only one term satisfies the definition of an arithmetic sequence by vacuity. It may feels weird, in the same way a function with a domain with only one elements is considered injective, but giving definitions that are more general often serves to avoid the necessity to exclude extreme cases when doing proofs about those concepts. – jjagmath Nov 26 '23 at 03:57

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Mathematics tends to prefer general definitions.

For example, a function $f: \mathbb{Z} \rightarrow X$ mapping from the integers is typically considered continuous (under the discrete topology). You might argue that such a function doesn't "feel" continuous, or that it's not "useful" to extend the definition in this way, but in reality, it's more work to gatekeep trivial objects from satisfying the definition at hand. You should only exclude edge cases if they simplify the theory you are working towards.

All of this is to say that there might be nothing wrong with allowing a single term sequence to constitute a finite sequence. If nothing else, it saves you from having to stipulate at what point something is special enough to "become" a finite sequence (is it 2 terms? 3 terms?).

As I post this, I see that @jjagmath has answered with a comment making a similar observation; apologies for the overlap

parsiad
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