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 2nd difference sequence (Posted on 2005-10-22)
While working with a non-zero sequence recently, I noticed that when I found the second difference of the sequence, the result was identical to the original sequence. Specifically, the first term of the 2nd difference sequence was the same as the first term of the original sequence. And so on.
What is the limit (as n goes to infinity), of the ratio of the n-th term to the previous term?

Part 2
Another sequence has the property that each term of the 2nd difference sequence is equal to the corresponding term of the original sequence multiplied by "k", where k is a positive real number, not necessarily an integer.
For the original sequence, what is the ratio (in the limit) of the n-th term to the previous term?

Definition of 1st difference sequence:
For sequence: a(1), a(2), ..., a(n),...
1st difference is: a(2)-a(1), a(3)-a(2), ... a(n+1)-a(n),...

 See The Solution Submitted by Larry Rating: 3.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Hmmm... (technical solution to part one, but probably not the one in mind) | Comment 1 of 11
For the first sequence, it could be 2^n: 2, 4, 8, 16, 32, ..., is identical to all its difference sequences--1st, 2nd, etc.  The ratio of terms is a constant 2, and is therefore the limit as well.  The question seems to imply, though, that the first differences are not the original sequence, and that the ratio of the second difference varies but approaches a value, so doesn't sound like this series.
 Posted by Charlie on 2005-10-23 14:13:25

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