= a and a2 and the 0 and b + 4c 0. It for all integers...

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= a and a2 and the 0 and b + 4c 0. It for all integers n 1 2. Consider the sequence a1, a2, a3, ... defined by some initial values a = recurrence relation an = b an1 + c An-2 for n 3 with 60 and c is a known fact (that you do not need to prove) that an = srn + tr and for some real numbers s and t, where r and 2 are the roots of the quadratic equation x-bx- c = = 0. (a) Prove that if then An = s. r+trn for all integers n 1 and for some real numbers s and t, where r and r are the roots of the quadratic equation x bx c = 0 An = = b. An1 + C An2 for all integers n 3. = = 1, a2 3, and an = (b) Consider the sequence defined by a an-12an-2 for n 3. Determine the values of a3, 04, 05, 06, and a closed form expression for an of the form an = srn + t rm. . (c) Consider the sequence defined by a = 1, a2 = -3, and an = 2an 15an-2 for n 3. Determine the values of a3, 4, 5, 6, and a closed form expression for an of the form An = s r + t r. (In this part, 71, 72, s and t might be complex numbers.) . (d) Consider the sequence defined by so = 1, 81 = 2, S2 = 9, and Sn - 38n-138n-2 - Sn-3 for n 3. Determine the values of S4, S5, S6, S7, and a closed form expression for sn. = a and a2 and the 0 and b + 4c 0. It for all integers n 1 2. Consider the sequence a1, a2, a3, ... defined by some initial values a = recurrence relation an = b an1 + c An-2 for n 3 with 60 and c is a known fact (that you do not need to prove) that an = srn + tr and for some real numbers s and t, where r and 2 are the roots of the quadratic equation x-bx- c = = 0. (a) Prove that if then An = s. r+trn for all integers n 1 and for some real numbers s and t, where r and r are the roots of the quadratic equation x bx c = 0 An = = b. An1 + C An2 for all integers n 3. = = 1, a2 3, and an = (b) Consider the sequence defined by a an-12an-2 for n 3. Determine the values of a3, 04, 05, 06, and a closed form expression for an of the form an = srn + t rm. . (c) Consider the sequence defined by a = 1, a2 = -3, and an = 2an 15an-2 for n 3. Determine the values of a3, 4, 5, 6, and a closed form expression for an of the form An = s r + t r. (In this part, 71, 72, s and t might be complex numbers.) . (d) Consider the sequence defined by so = 1, 81 = 2, S2 = 9, and Sn - 38n-138n-2 - Sn-3 for n 3. Determine the values of S4, S5, S6, S7, and a closed form expression for sn.