Quandaries
and Queries
Who is asking: Student
Level: Secondary
Question:
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;
and
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
Penny