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August 7th, 2017, 02:44 PM  #1 
Newbie Joined: Jul 2017 From: Pennsylvania Posts: 10 Thanks: 0  Question concerning infinite sequences
I just finished up our section on arithmetic and geometric sequences in class, both finite and infinite. Was pretty easy, but I have a question that google hasn't been terribly helpful in answering (I always seem to be the oddball, finding weird questions): Many of our problems started with a given sequence, such as: 1,4,7,... for example. We know the "..." means to infinity (and beyond!). So, me being me, wondered is "...,1,4,7,..." also a valid infinite sequence, and if it is, what then is the first term in that sequence (for purposes of calculating sums and nth terms)? If I use 1 as the first term, then sure, everything works in the positive direction, but what about the other way? Can I reverse the difference (make it negative) and "back" calculate for terms before 1, say for example what is the "a20" term? Is that "legal" in the math world? 
August 7th, 2017, 03:39 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,758 Thanks: 1409 
in the arithmetic sequence $\{... , 2, 1, 4, 7, ... \}$ 1 need not be $a_1$, which is why we may generally define the nth term of an arithmetic sequence as $a_n = a_1 + (n1)d$ so, if you prefer $a_{21} = 1$, then you can determine $a_1$ from there. 
August 7th, 2017, 03:58 PM  #3 
Newbie Joined: Jul 2017 From: Pennsylvania Posts: 10 Thanks: 0 
So ...,1,4,7,... is a valid sequence, and we can set a1 as any term in the sequence. That makes sense and is so incredibly simple. 
August 7th, 2017, 05:49 PM  #4  
Senior Member Joined: Aug 2012 Posts: 1,960 Thanks: 547  Quote:
..., 1, 4, 7, ... is not a sequence. It has no first element.  

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