Need help with Discrete Structures.

Tasks:

Write an iterative $C_{++}$function that inputs a nonnegative integer $n$ and returns

the $n$th Fibonacci number.

Write a recursive $C_{++}$function that inputs a nonnegative integer $n$ and returns

the $n$th Fibonacci number.

Compare the number of operations and time taken to compute Fibonacci

numbers recursively versus that needed to compute them iteratively.

Use the above functions to write a $C++$ program for solving each of the following

computational problems.

I. Find the exact value of $f_{100},f_{500},$ and $f_{1000},$ where $f_n$ is the $n$th Fibonacci

number. What are times taken to find out the exact values?

II$.$ Find the smallest Fibonacci number greater than $1,000,000$ and greater

than $1,000,000,000.$

III. Find as many prime Fibonacci numbers as you can. It is unknown whether

there are infinitely many of these. Find out the times taken to find first $10,$

$20,30,40$dots up to $200$ and draw a graph and see the pattern.

Report: Your report should address all of the below mentioned questions

Describe the Fibonacci series and write briefly what you have done in the

assignment.

What are the different skills, programming techniques have you used in order to

run the experiments?

What did you observe when you did the comparisons in Task $3$ and Task $4?$

Explain the graph that you have drawn from Task $4.$III?

List at least three different applications of the Fibonacci numbers to sciences and

describe one of them in details. Think of situation or a real world problem where

you can apply concept of Fibonacci numbers to solve it$.$ Explain?

Write a paragraph, explaining what you have done in this assignment. What were

the challenges you have faced when solving these problems. How can you

improve the programs you have written to solve these problems?