In the following list, select all the correct expressions if G(x) is the generatingfunction of the sequence {a}2o which satisfies the recursive relationak =2ak-1 + 5ak-2 +3k-1 for every integer .k > 2.G(x) = a + a₁x + 2x²₁ ax² +5%₁ akxk+² + xΣk=1□G(x) = a + a₁x+2Σ2 ak-1x² +5%-2 ak-200² +22 3k-1pk□ G(x) = aG(x) = a00 3kk1+ a₁x + 2x [G(x) − ao] + 5x²G(x) + x = I1-3+ a₁ + 2 Σ ₁ akx²+¹ +5Σakx²+² +Σ1 3kk+1700G(x) = a + a₁x + 2G(x) + 5xG(x) + x- I11-32

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Answered: In the following list, select all the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(Recursively-Defined Sequences). Consider the function f(x) = x + exp(4.73x). (i) Use a scientific calculator to generate the terms p2, P3, ... , Pg of the sequence (pn) recursively defined by Рп-1 — Рп-2 Pn = Pn-1 - f(Pn-1) f(Pn-1) - f(Pn-2) for all n > 2 whose initial terms are given by Po = -0.6 and p1 = -0.5. Before entering each of the terms pp in the corresponding answer field below, please round it to nine decimal places: P2 = P3 = P4 = P5 = P6 = P7 = Pg =
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Suppose we have a recursive sequence ƒ1, ƒ2, ƒ3, . . .. For the purposes of this problem, it does not matter exactly how the fi are defined, only that they are recursively defined. = i=1 For integer n ≥ 1, let P(n) be the predicate that Σï-1 fi = 2fn+2 − 3. Don't worry about whether this predicate "makes sense"; we haven't defined the fi so you won't be able to "make sense" of the P(n). It's not important for this problem. Consider a proof by induction that Vn ≥ 1: P(n). Suppose we've gotten to the inductive step, and suppose that the first steps of the inductive step are k+1 k+1 Σ1 fi = f1 + f2 + Σh fi = f₁ + f2 + Σkt k+1, i=3 (fi−1 + fi−2) = f₁ + f2 + ¹ fi- 1 + k fi 2 f1 k+1 k+1 k = f1 + f2 + Σh_2 fi + [h=1² fi i=2 = f2 + Σ²²₁ fi+fi i=1 i=1 True or false: Based on the information given, we will need strong induction for this proof. True False

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