3.Use strong induction to prove that C(n)= 2" + 3 is a solution to the recurrenceC(0) = 4, C(1) = 5, and, for all ne Z+, n>1C(n) = 3. C(n - 1) – 2. C(n – 2).%3D

Given, C(0)=4, C(1)=5, and C(n)=3. C(n-1)-2.C(n-2) for all n∈ℤ+, n>1 To prove that C(n)=2n+3…

Answered: 3. Use strong induction to prove that…

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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1. How to solve the following recurrence using backward substitution. T(n) = 2 * T (n/2) + c, forn > 1 T(1) = a O (n²) o (ni ) O(n log n) O (2")
Suppose we have the recurrence relation if n = 1) if n = 2 (W (n – 2) + 2W(n – 1) if n > 2) 1 W (n) : 3 Use induction to prove all resulting values of W (n) are odd for n 2 1.
2. Use the recursion tree method or the telescopic method to solve the recurrence relation f(0) = 7 and, for all n e Z+ f(n) = f(n – 1) – 2n.
8. Suppose that we have a linear homogeneous recurrence relation of degree 6 with constant coefficients and that its characteristic equation is given by (r – 3)³(r – 2)²(r – 1) = 0. Write down the general solution a, for the recurrence relation.
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Consider the following linear non-homogeneous recurrence relation where go = 1, g1 = 3 and gn†1=-9n + 6gn-1 - 3 for n > 1. The reduced closed form of gn can be represented as gn = (Px p" + Q x q" + C') where p and q are the solutions of the characteristic equation of the recurrence. Given that, p is smaller than q. Here, P,Q,C, A are all integers. What is the value of p? What is the value of P?? What is the value of C? What is the value of A?
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
7. Consider the second-order homogeneous recurrence relation an 5an-1 6an-2 with initial conditions ao : 0, a1 = 5. (a) Find the next three terms of the sequence. (b) Find the general solution. (c) Find the unique solution with the given initial conditions.
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3. Use strong induction to prove that C(n) = 2" +3 is a solution to the recurrence C(0) = 4, C(1) = 5, and, for all n E Zt, n > 1 C(n) = 3. C(n – 1) – 2. C(n – 2).