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),...

(In reply to re(4): Solution? ... and a suggestion by bernie)

bernie's solution looks good. 

The "leap" assumes that the sequence of ratios converges i.e. lim R(n)/R(n-1) < 1.  I have not been able to prove this analytically  (Have bernie or oscar?) but have not found any exceptions.

1+sqrt(k) gives limit for sqrt(k)>= 0;

1- sqrt(k) gives limit for sqrt(k)<0;

 

Posted by goFish on 2005-10-24 10:22:21