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August 7th, 2017, 02:44 PM   #1
EDL
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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?
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August 7th, 2017, 03:39 PM   #2
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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

$a_n = a_1 + (n-1)d$

so, if you prefer $a_{21} = 1$, then you can determine $a_1$ from there.
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August 7th, 2017, 03:58 PM   #3
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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.
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August 7th, 2017, 05:49 PM   #4
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Quote:
Originally Posted by EDL View Post
So ...,1,4,7,... is a valid sequence, and we can set a1 as any term in the sequence.
No that is not correct. An infinite sequence is by definition a function from the natural numbers 1, 2, 3, ... to your target set. So there must be a first element of the sequence, a second, a third, etc.

..., 1, 4, 7, ... is not a sequence. It has no first element.
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