The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding numbers. Here's how a Fibonacci series function works:

Functionality:

The function typically takes an integer n as input, representing the number of elements desired in the Fibonacci sequence.

Steps:

Base Cases: The function usually defines two base cases to handle the starting values of the sequence, which are typically $0$ and $1.$

Recursive or Iterative approach: The function implements either a recursive or iterative approach to calculate the remaining Fibonacci numbers.

Recursive approach: Defines a function that calls itself with n$-1$ and n$-2$ as input to calculate the previous two numbers and then returns their sum. This approach can be less efficient for larger values of n due to repeated function calls.

Iterative approach: Uses a loop to iterate through the desired number of elements, keeping track of the previous two numbers and calculating the next one based on their sum. This approach is generally more efficient for larger sequences.

Return the Sequence: The function returns the complete Fibonacci sequence up to the n$-$th element $($often as a list or array$)$ or just the n$-$th Fibonacci number

here's the Fibonacci sequence function in pseudocode:

function fibonacci$($n$)$

$//$ Base cases

if n $<=1$

return n

$//$ Initialize variables to store the previous two numbers

prev$\_1$ :$=0$

prev$\_2$ :$=1$

$//$ Loop to calculate remaining Fibonacci numbers

for i from $2$ to n

current :$=$ prev$\_1+$ prev$\_2$

prev$\_1$ :$=$ prev$\_2$

prev$\_2$ :$=$ current

$//$ Return the nth Fibonacci number

return current

end function

$1-$ Implement $$fibonacci$$ in at least two programming languages one interpreted the other compiled.

$2-$ run each implementation using different inputs $($n$).$

Report the time needed for each run. $($use graphs and tables$)$

Compare the performance.

Discuss and conclude the timing results.

The deliverables :

Code:

$*$ Two correctly implemented versions of the fibonacci$($n$)$ function:

$*$ One in an interpreted language $($e$.$g$.,$ Python$)$

$*$ One in a compiled language $($e$.$g$.,$ C$++)$

$*$ The code should be well$-$formatted, indented, and include comments explaining the logic.

Analysis and Results:

$*$ A table or list showing the chosen input values $($n$)$ for running the Fibonacci functions.

$*$ Corresponding output $($resulting Fibonacci numbers$)$ for each input value.

$*$ A table or chart displaying the execution time required for each run $($interpreted vs$.$ compiled languages$)$ using different input values.

Report:

$*$ A discussion comparing the performance of the interpreted and compiled languages based on the execution time results.

$*$ An explanation for the observed performance variations.

$*$ A conclusion summarizing the key takeaways from the assignment, including the impact of interpreted vs compiled languages and any optimizations explored.

Optional Bonus Deliverables:

$*$ Code demonstrating optimizations made to improve the performance of the function $($e$.$g$.,$ memoization in a recursive approach$).$

$*$ A graph or chart visualizing the execution time data for different input values.

$*$ A discussion of potential future improvements or further exploration related to the Fibonacci sequence function or performance analysis.

Overall Presentation:

$*$ The assignment should be well$-$written, organized, and use proper grammar and formatting.

Assignment evaluation methodology $($for TA aattention$)$

Implementation $|40|$

$*$ Correctness $(20$ points$)$: The provided Fibonacci function $($fibonacci$($n$))$ must be correctly implemented in at least two programming languages: one interpreted $($e$.$g$.,$ Python$)$ and one compiled $($e$.$g$.,$ C$++).$ Each implementation $(10$ points each$)$ should accurately calculate the Fibonacci sequence for different input values.

$*$ Code Clarity $(10$ points$)$: The code should be well$-$formatted, indented, and include comments explaining the logic behind each step $($variable names, loops, calculations$).$

$*$ Efficiency Considerations $(10$ points$)$: $($Bonus$)$ If applicable to the chosen languages, discuss any optimizations made to improve the performance of the function, especially for larger input values $($e$.$g$.,$ using memoization in a recursive approach$).$

Analysis and Results $|40|$

$*$ Running the Implementations $(20$ points$)$: The assignment should demonstrate running each implementation $($fibonacci$($n$))$ with different input values $($n$).$ Include a table or list showing the chosen input values and the corresponding output $($resulting Fibonacci numbers$).$