Example 4: Consider the following version of the Fibonacci sequence starting from Fo= 0 and defined by: Fo=0 F₁ = 1 } Fibonacci Sequence Fn+2Fn+1+ F₁; n ≥ 0: Prove the following identity, for any fixed k ≥ 1 and all n ≥ 0. Fn+kFkFn+1 + Fx-1Fn- The Fibonacci Sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... Assertion: Fn+k=FxFn+1 + Fk-1Fn: for any fixed k ≥ 1 and all n ≥ 0 k-1, n=0 F₁ F₁F₁ + FoFo = 1 k=1, n=1 F₁₂ F₁F₂ + FoF₁ = ? 1 = 1*1 + 0*1 = 1 k-5, n=4 F, F5F5 + F4F4 = 5*5 +3*3 = 34 Assumption for pair (k,n: k≥ 1, n ≥ 0) assertion is valid. Is this identity true for n+1, k+1? Fn+k+2= Fk+1Fn+2 + FkFn+1 ?? True??

Example 4: Consider the following version of the Fibonacci sequence starting from Fo= 0 and defined by: Fo=0 F₁ = 1 } Fibonacci Sequence Fn+2Fn+1+ F₁; n ≥ 0: Prove the following identity, for any fixed k ≥ 1 and all n ≥ 0. Fn+kFkFn+1 + Fx-1Fn- The Fibonacci Sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... Assertion: Fn+k=FxFn+1 + Fk-1Fn: for any fixed k ≥ 1 and all n ≥ 0 k-1, n=0 F₁ F₁F₁ + FoFo = 1 k=1, n=1 F₁₂ F₁F₂ + FoF₁ = ? 1 = 11 + 01 = 1 k-5, n=4 F, F5F5 + F4F4 = 55 +33 = 34 Assumption for pair (k,n: k≥ 1, n ≥ 0) assertion is valid. Is this identity true for n+1, k+1? Fn+k+2= Fk+1Fn+2 + FkFn+1 ?? True??

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

See similar textbooks

Similar questions

Recurrence relations: Master theorem for decreasing functions T(n) = {₁T(n- aT(n −b) + f(n), if n = 0 if n > 0 f(n) = nd What is T(n)?
The Legendre Polynomials are a sequence of polynomials with applications in numerical analysis. They can be defined by the following recurrence relation: for any natural number n > 1. Po(x) = 1, P₁(x) = x, Pn(x) = − ((2n − 1)x Pn-1(x) — (n − 1) Pn-2(x)), n Write a function P(n,x) that returns the value of the nth Legendre polynomial evaluated at the point x. Hint: It may be helpful to define P(n,x) recursively.
Solve the recurrence relation without master's theorem : T(n)= 7T(n/2)+ n^2
Mathematical Induction: Binet's formula is a closed form expression for Fibonacci numbers. Prove that binet(n) =fib(n). Hint: observe that p? = p +1 and ² = b + 1. function fib(n) is function binet(n) is match n with let case 0 → 0 2 case 1 1 otherwise in L fib(n – 1) + fib(n – 2) V5
By using recursive approach perform the following first 3 problems by using functions in Java Anlyse their working mechanisms and check their correctness Comment on their T(n)
Suppose that f(n) satisﬁes the divide-and-conquer recurrence relation f(n) = 3f(n/4)+n2/8 with f(1) = 2. What is f(64)?
8. Solve the following recurrence relations. (16 points) a. T(n) = 4T(n-1) + 1 for n > 1; T(1) = 1; b. T(n) = T(n/2) + 3; T(1) = 1; Here you can assume that n=2k for some integer k.
Let T(n) be defined by the recurrence relation T(1) = 1, and T(n) = 32T(n/2) + nk for all n > 1. Determine the integer k for which T(n) = 0(n³ log n).
Q/ Given a sequence x(n), n from 0 to 3, where x(0)=-2, x(1)=4, x(2)=0 and x(3)=-5 Evalute its DFT X(K)
Let T(n) be defined by the following recurrence relation T(0) = T(1) = 1 T(n) = T(n-1) +T(n − 2) +1 for n ≥2 Show that T(n) = 2Fn1 for n ≥ 0, where Fn is the nth Fibonacci number, i.e., Fo=F₁ = 1; Fn = Fn-1 + Fn-2 for n ≥ 2.
Exercise • Exercise Given a (7 , 4) non-systematic code 0 1 0 0 1 0 1 1 1 0 0 1 1 0 G= 0 1 1 0 1 1 0 0 0 1 10 1 Solve for s if (a) r = [1101011], and (b) r = [1010111]
Determine P(A x B) – (A x B) where A = {a} and B = {1, 2}.