2. Let (F(n))n>o be a sequence of real numbers and let k € N.(a) Explain what it means for (F(n))n>0 to satisfy a (k +1)-term recurrence relation withconstant coefficients.(b) Write down the generating function for (F(n))n>0-(c) Suppose the sequence (F(n))n>o has the following initial terms F(0) = 1, F(1) = 4, andsatisfies the recurrence relationF(n) = 2F(n – 1) + 8F(n – 2) for n> 2.|Using any approach, find a closed form for F(n) for all n > 0.

Given that, (Fn)n≥0 is the real numbers sequence and let k∈N (a) A sequence (Fn)n≥0 of real numbers…

Answered: Let (F(n))n>0 be a sequence of real…

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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Calculate the result of f(n) using recursion. Depict the recursive tree and demonstrate the result calculation for the following recursive functions. f=) + 1, if n is even iii) f(n) = , f(1) = 1, calculate the result for N = 80. f () + 1,if n is odd. fE) + n, if n is even iv) f(n) = f(1) = 1, calculate the result for N = 80. f () - 5, if n is odd.
Find the sequence {ax}k2o for the generating function G(x) = (x + 2)².
In the following list, select all the correct expressions if G(x) is the generating function of the sequence {a}2o which satisfies the recursive relation ak = 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) = a G(x) = a 00 3kk 1 + a₁x + 2x [G(x) − ao] + 5x²G(x) + x = I 1-3 + a₁ + 2 Σ ₁ akx²+¹ +5Σakx²+² +Σ1 3kk+1 700 G(x) = a + a₁x + 2G(x) + 5xG(x) + x- I 1 1-32
Consider the following recursive function: if n = 0 f(n) = {2. f(n-1)-5n+3 ifn≥1 Prove that the closed form is f(n) = 2" +5n+7.
For each of these generating functions, provide a closed formula for the sequence it determines. a) (3x – 4)3 с) 1/(1 - 5х). e) x + 3x +7+(1/(1 –x²)) г) (1- х) -х3- х? —х-1 g) x/(i – x)? b) ( + 1)3 d) x/(1+3x) h) 2e*
The closed form solution to the recurrence relation a(n) = a(n-1) + 4 with a(0) = 3 is (A) a (n+1)= a(n) - 3 (B) a(n) = 4 + 3n (C) a(n) = 3 + 4n (D) a(n) = 4n-3 (E) None of the above.
d [erf (x)] e* 2 (-prore) using sequences
(a) Male honeybees are from single parent families; they are from unfertilized eggs so only have mothers. Female honeybees are from two parent families; they are from fertilized eggs so have both mothers and fathers. Let Bn bethe number of nth generation ancestors of a male honeybee. For example: B1 = 1 (itself), B2 = 1 (its mother), B3 = 2 (its grand-father and grand-mother), etc. Find a recurrence relation for Bn. [Assume that no honeybee appears more than once in the ancestral tree; that is, each honeybee has at most one child in the tree.](b) Prove that if n is divisible by four then Bn is divisible by three.
Let an be the number of strings (i.e. sequences of letters, non-sense words) of length n that can be formed using the four letters Q,X,T and Z. (a)Find an explicit formula for an (b)Find a recurrence relation that is satisfied by the numbers an. (c) Find a rational generating function (ratio of two polynomials) that is a generating function for the numbers an.
4. Consider the following sequence: F(1) = 8 F(2) = 16 F(n) = 6F(n − 1) - 5F(n 2) for n ≥ 3 indicate which kind of recurrence relation it is (linear/not-linear, first/second/n-th order, homogeneous or not, constant/no-constant coefficients, divide- and-conquer): Find the closed-form expression of the recursive sequence
Consider the recurrence relation: T(n) = { a, if n == 1 { b + cN + T(n-1), if N >= 2 Determine the closed form. Show your work.
Given the recursive formula, ƒ (n) = f (n − 1) + ƒ (n − 2), n ≥ 3, and f(1) = f (2) = 1. What is f (8)?

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