Question

Honeybee reproduction is an interesting phenomenon. When a queen bee lays an egg, the egg does not need to be fertilized by a male honeybee to hatch. The egg will develop into a male or a female honeybee depending on whether the egg is fertilized by a male honeybee. The outcome is determined by the following rules:
· If an egg is not fertilized, it hatches a drone, or male, honeybee.
· If an egg is fertilized, it hatches a worker, or female, honeybee.

a. Based on this information, how many parents will a drone have?
b. How many grandparents will a drone have?
c. How many great-grandparents, great-great-grandparents, and great-great-great-grandparents will a drone have?
d. Write the numbers together, and determine what type of sequence they form.
e. Write the sequence using a recursive function
f. Use the Internet to research and list at least three other real-world situations that you can model with the Fibonacci sequence.
g. In your research, what famous number or ratio related to the Fibonacci sequence did you come across?

Answer 1

According to the problem, Drones have just mother & no fathers.
Having said that it become easy to solve the problem:

a. Based on this information, how many parents will a drone have? 1

b. How many grandparents will a drone have? 2

c. 1)How many great-grandparents, 3
    2)great-great-grandparents, 5
    3)great-great-great-grandparents will a drone have, 8

d. Write the numbers together, and determine what type of sequence they form:
1,2, 3, 5, 8 (notice that the 3rd term = the sum of the preceding numbers 3=1+2,  5=3+2, 8= 5+3) So it's a Fibonacci sequence

e. Write the sequence using a recursive function: a(n) = a(n-2)+a(n-1)

f. Use the Internet to research: Do it & you will find several related facts

g. In your research, what famous number ...
    Let's take again the Fibonacci sequence:
     1, 2, 3, 5, 8, 13, 21, 34, ...
     Now let's evaluate the ratio between 2 consecutive number:
      3/2 =1.5
      5/3=  1.666
      8/5 =  1.6
      13/8 = 1.625
      21/13 =1.615
      34/21 = 1.619
This ratio tends to be 1.618 which is calle the Golden Ratio