Let n=0 be a natural number. A sequence of numbers is briefly called a sequence “Fn” if n different numbers z1, z2, ..., zn exist so that the following conditions are fulfilled:
(1) Each term of the sequence is one of the numbers z1, z2, ..., zn.
(2) Each of the numbers z1, z2, ..., zn occurs at least once in the sequence.
(3) Any two immediately consecutive members of the sequence are different numbers.
(4) No subsequence of the sequence has the form {a,b,a,b} with a=b.Note: A subsequence of a given sequence {x1,x2,x3,...} or {x1,x2,x3,...,xs} is called any sequence of the form {xm1,xm2,xm3,...} or {xm1,xm2,xm3,...,xmt} with natural numbers m1<m2<m3<...Answer the following questions:
a) Given n, are there sequences Fn of arbitrarily long length?
b) If question (a) is answered in the negative for an n:
What is the largest possible number of terms that a sequence Fn can have (given n)? combinatoricsalgebraSequence