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

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Eight puzzle problem

The N-puzzle problem contains N tiles (N+1 including empty tiles) where the value of N can be 8, 15, 24, etc. The puzzle is divided into the square root of (N+1) rows and the square root of (N+1) columns. These problems have an initial state or configuration and a goal state or configuration. The puzzle can be solved by shifting the tiles one by one in the empty space. The objective is to place the numbers on the tiles to match the final state using the empty space. The other name for the N-puzzle problem is the sliding puzzle.

Question

Example 4:
Consider the following version of the Fibonacci sequence starting from F, = 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+kFxFn+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
F₁ F₂ + 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=Fx+1Fn+2 + FkFn+1 ?? True??

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