Need help with Question #4.Question #3 has been provided as a sample reference with work. Fn(Fn+1Fn-1.3. Let Fn denote the nth Fibonacci number. Let.Show that A" =(Hint: use induction).Given,By Induction:For n=1:A =F1+1F1A'=To prove that,F1F1-1- ( )- (; :)-^Fa+1(F, Fn-1F2 F1F, FoA"{:: F2 = 1, F1 = 1, and Fo = 0}Therefore, A" is true for n = 1.For n=k:Assume that the given statement is true for n= k. Then we haveAt = ("- (*)´F+1Now to prove that The given statement is true forn = k+1 i.e.F142 Fr+1Ak+1L.H.S.Ak+l = Ak.A- (":F+1F Fx-1{:: by u sin g (1), and A is Given}- ( )- (Fr+1 + Fr Fk+1 +0Fz + Fx-1 Fx + 0Fr+2 F+1= R. H.SF+1Thus, A" is true for n = k + 1.Hence, by the principle of mathematical induction, the given statement istrue for all natural numbers (n eN).. 4. Use the equality proven in question 3 to design an algorithm to compute the nth Fibonaccinumber in O(log n) time.

Answer: I have done code and also I have attached code code screenshot as well as output in C++…

Answered: 4. Use the equality proven in question…

Engineering

Computer Science

4. Use the equality proven in question 3 to design an algorithm to compute the nth Fibonacci number in O(log n) time.

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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Question

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Need help with Question #4.

Question #3 has been provided as a sample reference with work.

Fn
(Fn+1
Fn-1.
3. Let Fn denote the nth Fibonacci number. Let.
Show that A" =
(Hint: use induction).
Given,
By Induction:
For n=1:
A =
F1+1
F1
A'=
To prove that,
F1
F1-1
- ( )
- (; :)-^
Fa+1
(F, Fn-1
F2 F1
F, Fo
A"
{:: F2 = 1, F1 = 1, and Fo = 0}
Therefore, A" is true for n = 1.
For n=k:
Assume that the given statement is true for n= k. Then we have
At = ("
- (*)
´F+1
Now to prove that The given statement is true forn = k+1 i.e.
F142 Fr+1
Ak+1
L.H.S.
Ak+l = Ak.A
- (":
F+1
F Fx-1
{:: by u sin g (1), and A is Given}
- ( )
- (
Fr+1 + Fr Fk+1 +0
Fz + Fx-1 Fx + 0
Fr+2 F+1
= R. H.S
F+1
Thus, A" is true for n = k + 1.
Hence, by the principle of mathematical induction, the given statement is
true for all natural numbers (n eN)..

4. Use the equality proven in question 3 to design an algorithm to compute the nth Fibonacci
number in O(log n) time.

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