Problem 5. Consider the vector space V of all infinite sequences of real numbers (a0, a1, a2, .,). Fix a real number b. .,) such that an+2 = an+1 + ban for (a) Consider the subset W₁₂ of all sequences (a0, a1, a2, . . ., all n ≥0. Prove that W₁ is a subspace of V. (b) Consider the sequence (Sn) E W₁ whose first two terms are so the first 8 terms of this sequence. (c) Determine the dimension of W₁, with proof. = 0 and s₁ = 1. Write out (d) For which values of b does W₁ contain a sequence of the form (r, r², p³, ...) for some r 0? Such a sequence is called a geometric sequence. (e) When 6 = 6, find all non-zero values of r such that the geometric sequence (r, r², r³, ...) E W6- (f) Find a basis for W6 consisting of geometric sequences. (g) Find a closed form for the n-th term of the sequence (0, 1, 1, 7, 13, 55, ...) Є W6. [Hint: Use your basis from (f).] Bonus Problem. Use the ideas of problem 5 to come up with a closed form for the n-th Fibonacci number (the sequence in b).