Deliverable For each of the problems shown below, clearly define what the output

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Deliverable

For each of the problems shown below, clearly define what the output, input and processing tasks should be to meet the requirements you have been given.

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PROBLEMS

(10 pts) A program determines the position n of a particular number in the Fibonacci sequence. Fibonacci numbers Fn are defined by the recursive relation[1] below. It is assumed that the Fn provided is a valid Fibonacci number.

where the initial seeds are F0 = 0 and F1 = 1. The following table shows the position n of the first six numbers Fn in the Fibonacci sequence (including the seeds 0 and 1);

Position (n)

0

1

2

3

4

5

Fibonacci number (Fn)

0

1

1

2

3

5

The formula to calculate the position n of a Fibonacci number Fn in the Fibonacci sequence is:

                  is the golden ratio.

     

Equation (1) will be evaluated twice (i.e., once with +4, and once with -4) and only integer solutions can be used to represent the position n of a Fibonacci number Fn. (Hint: The logarithm change-of-base formula can help simplify complicated logarithm functions.)

[1] https://en.wikipedia.org/wiki/Fibonacci_number

Position (n)

0

1

2

3

4

5

Fibonacci number (Fn)

0

1

1

2

3

5

A program determines the position n of a particular number in the Fibonacci sequence. Fibonacci numbers Fn are defined by the recursive relation[1] below. It is assumed that the Fn provided is a valid Fibonacci number. where the initial seeds are F_0 = 0 and F_1 = 1. The following table shows the position n of the first six numbers F_n in the Fibonacci sequence (including the seeds 0 and 1): The formula to calculate the position n of a Fibonacci number F_n in the Fibonacci sequence is: n = log_phi (F_n Squareroot 5 + Squareroot 5 F_n ^2 plusminus 4/2) is the golden ratio. phi = 1 + Squareroot 5/2 almostequalto 1.6180339887 .. Equation (1) will be evaluated twice (i.e., once with +4, and once with -4) and only integer solutions can be used to represent the position n of a Fibonacci number F_n.

Explanation / Answer

The program is given below:

#include<stdio.h>
#include<math.h>
int position(int f)//function to calculate position
{
float phi=(1+sqrt(5))/2;//golden number
float x1=(f*sqrt(5)+sqrt(5*f*f+4))/2;
float x2=(f*sqrt(5)+sqrt(5*f*f-4))/2;
float p1,p2;
p1=log(x1)/log(phi);//Equation evaluated using +4
p2=log(x2)/log(phi);//Equation evaluated using -4
return (p1==ceil(p1)?p1:p2);//returning the integer of the two values
}
void main()
{
int f;
printf("Enter the Fibonacci number:");
scanf("%d",&f);//Fn is entered
printf("The position of %d is %d",f,position(f));//Position of Fn is printed using the function
}

A sample output for this program is given below:

Enter the Fibonacci number:13
The position of 13 is 7

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