(*10 points*)
**I. 96.** The first *n* terms of a sequence do not specify the rest: in general, any number can stand at the (*n*+1)-th place.
We now give the following rule to determine the (*n*+1)-th term (supposing that *n* \(\displaystyle \ge\)2 is an integer and the first *n* terms of the sequence are already given):
The difference of consecutive terms are formed (that is the term with smaller index is subtracted from the one with larger index). Then we do the same operation on the difference sequence just obtained, and repeat the procedure again. After a few steps we arrive at a constant sequence with all terms equal: this will surely happen at most in the (*n*-1)-th step, since this time the sequence consists of only one term. For two examples, see the numbers in bold face in the table:
Your task is to find the (*n*+1)-th term of the original sequence such that forming difference sequences from the beginning will yield the same constant sequence. (See the thinner numbers at the rightmost columns in the example.)
Prepare a sheet (i96.xls) that constructs the (*n*+1)-th term of the sequence, if the first *n* terms are given, using the rule above.
Your sheet (i96.xls) is to be submitted. Remarks about its usage may be entered into cell `A1`. |