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A sequence is just an infinite list of numbers (say real numbers, we often denote these...

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Advance Math

A sequence is just an infinite list of numbers (say realnumbers, we often denote these by a0,a1,a2,a3,a4,.....,ak,..... sothat ak denotes the k-th term in the sequence. It is not hard tosee that the set of all sequences, which we will call S, is avector space.

a) Consider the subset, F, of all sequences, S, which satisfy:?k ? 2,a(sub)k = a(sub)k?1 + a(sub)k?2. Prove that F is a vectorsubspace of S.

b) Prove that if 10,a1,a2,a3,.... is a sequence if F for whicha0=a1=0 then the sequence is the zero sequence, that is ?k ?0,a(sub)k = 0

c) Prove that the vector space F has dimension at most 2.

d) Prove that the sequences given by x(sub)k = ((1+root(5))/2)^kand y(sub)k = ((1-root(5))/2)^k are both elements in F and arelinearly independent.

e) Consider the sequence defined recursively by a0=0, a1=1 ?k> 1; ak = ak?1 + ak?2 , express this sequence an as a linearcombination of xn and yn.

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