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

A sequence is just an infinite list of numbers (say real numbers, we often denote these by a0,a1,a2,a3,a4,.....,ak,..... so that ak denotes the k-th term in the sequence. It is not hard to see that the set of all sequences, which we will call S, is a vector 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 vector subspace of S.

b) Prove that if 10,a1,a2,a3,.... is a sequence if F for which a0=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)^k and y(sub)k = ((1-root(5))/2)^k are both elements in F and are linearly 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 linear combination of xn and yn.

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Expert Solution

Colby Messinger answered 6 months ago

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