3.Consider the recurrence relation an = an-2 with initial conditions ao2 and a1-1.Use Strong Mathematical Induction to show13(-1)"2(for all n > 0)1"nAnis a closed form of the recurrence relation above.

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Answered: 3. Consider the recurrence relation an…

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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2. The recurrence relation а, — ба — 5а,, for n 2 2 п-2 satisfies a, = -1 and a, = 7. What is a,? Find the solution to the recurrence relation.
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Is it possible to use recurrence relations in functions?. If so give an example otherwisegive a valid reason. And also find a correct explicit formulae for the recurrence relation given in the image below
nan (a – 3)n-1 + x Σ an (x - 3)" = 0, - n=0 then the recurrence relation can be written as (n + 1)an+1 – an−1 + 3an = 0, (n + 1)an+1 + an−1 + an=0, – (n+1)an+1 + an−1 + 3an = 0, η>1 (n + 2)an+2 + an−1 + an = 0, (n + 2)an+2 + an−1 + an = 0, - Q6. 11 Σ n=1 = (a) αι = =do, (b) αι = 3a0 , (c) a1 = - –3ao, (d) a1 = do , (e) a1 = =do , n21 n>1 η21 η>1
Find the Recurrence Relation: * x²y" –`y = 0; at x, = 1
2. (a) Let (F(n)n>o be a sequence of real numbers and let k E N. Explain what it means for (F(n))n>o to satisfy a (k + 1)-term recurrence relation with constant coefficients. (b) Suppose (F(n))n>0 satisfies the following recurrence relation: F(n) = aF(n – 1) for n>1 and F(0) = b, where a, b e N. Compute the values of F(1), F(2), and then find a general formula for F(n) for arbitrary n > 0. (c) If (F(n))n>0 is a sequence, write down the generating function for (F(n))n>0- (d) A sequence (F(n))n>0 has the following initial terms F(0) = 2, F(1) = 4, F(2) = 10 and satisfies the recurrence relation F(n) = 3F(n – 1) – 2 for n > 1. 1 (i) Show that this sequence also satisfies the relation F(n) = 4F(n – 1) – 3F(n – 2) for n> 2. (ii) Using the generating functions approach, find a closed form for F(n) for all n> 0.
Use generating functions to solve the following non-homogenous recurrence relation: an = an-1 + n(n-1), where a0 = 3.
plz provide handwritten answer asap for ques 1
A sequence is given by the recurrence relation Xn+1 = Xn + 3(n + 1) for n > 1, and the initial value x1 = = 1. (a) Showing your working, use the recurrence relation to compute n for n = 1, 2, 3 and 4. 3n2 + 3n – 4 (b) Prove for all n >1 that xn = 2
4. Consider this recurrence relations an an-1 + 2an-2. (a) What sequence do you get if the initial conditions are ao=3, a₁=7? Give a closed formula for this sequence. (b) What sequence do you get if the initial conditions are ao=2, a₁=7? Give a closed formula. (c) What if ao 2 and a₁=5? Find a closed formula.
What is the solution to the recurrence relation: Tn=5nTn-1+6 with T0=3?

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3. Consider the recurrence relation an = an-2 with initial conditions ao = 2 and a1 = -1. Use Strong Mathematical Induction to show 1 1" (-1)" (for all n > 0) an = 2 is a closed form of the recurrence relation above.