Question

You will be exploring the Fibonacci sequence through programming. Complete the following tasks:

1. Research and take note of the recursive formula F(n) that can be used to define the Fibonacci sequence.
2. Design a simple program, using pseudocode, to implement the recursive formula you found in part (a) to compute numbers in the Fibonacci sequence. Describe in detail how your program implements the recursive formula. You may find it useful to discuss how it through a concrete example such as F(8) = 21.
3. Determine the number of times your program computes F(1) for each time F(5) is computed.
4. Discuss any issues you find with your program and what reasoning there may be

Answer 1

The recursive formula F(n) that can be used to define the Fibonacci sequence.
F(n) = F(n-1) + F(n-2)   when n >= 2
F(n) = 0   when n=0
F(n) = 1   when n=1

Pseudocode
Fibonacci(n):
   if(n == 0) return 0;
   if(n == 1) return 1;
   return Fibonacci(n-1) + Fibonacci(n-2)

calculation F(8)
F(8) = F(7) + F(6)
F(8) = F(6) + F(5) + F(5) + F(4) = F(6) + 2*F(5) + F(4)
F(8) = F(5) + F(4) + 2*F(5) + F(4) = 3*F(5) + 2*F(4)
F(8) = 3*(F(4) + F(3)) + 2*(F(3) + F(2))
F(8) = 3*(F(3) + F(2) + F(3)) + 2*(F(2) + F(1) + F(2))
F(8) = 3*(2*F(3) + F(2)) + 2*(2*F(2) + F(1))
F(8) = 3*(2*(F(2) + F(1)) + F(2)) + 2*(2*(F(1) + F(0)) + F(1))
F(8) = 3*(3*F(2) + 2*F(1)) + 2*(3*F(1) + 2*F(0))
F(8) = 3*(3*(F(1) + F(0)) + 2*F(1)) + 2*(3*F(1) + 2*F(0))
F(8) = (15*F(1) + 9*F(0)) + (6*F(1) + 4*F(0))
F(8) = 15*1 + 9*0 + 6*1 + 4*0
F(8) = 15 + 6 = 21

Number of times F(1) executed for each F(5) execution = 5 times

We can observe that this implementation does a lot of repeated work. For calculating F(5) , F(1) executed 5 times.