Quandaries and Queries


Who is asking: Student
Level: Secondary

Is it possible to have two formulas that define sequences that agree on the first four terms but not the rest?



Yes you can have different sequences agreeing in the first n terms but not on the rest for any n, and 4 in particular. You simply need two different n-step recurrences with the same initial conditions.

For example if n = 2:

Define a(n) = a(n-1)+a(n-2) with a(1) = a(2) = 1;
define b(n) = b(n-1)+3b(n-2) with b(1) = b(2) = 1,
then {a(n)} = {1,1,2,3,5,8,13, ...} and b(n) = {1,1,4,7,19,40, ...], etc



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