In this project, you will study some interesting numbers that arise frequently in nature: the Fibonacci numbers and the golden ratio. A great deal of information about these numbers can be found by searching the internet or your local library.

1. Suppose that a cow produces its first female calf at an age of two years, and thereafter produces a single female calf each year. Assuming none die, how many female cows will there be after 12 years? (Hint: Let F_n be the number of female cows after n years. Then it must be true that F₁ = 1, F₂ = 1, and F₃ = 2. Continue by finding the pattern.)
2. Determine the relationship among any three consecutive terms of the sequence discussed above. Find the first 25 terms of this sequence.
3. This sequence is named after the mathematician Fibonacci. Write a biographical paragraph about Fibonacci.
4. Find at least one other application where the Fibonacci sequence arises.
5. For each of the 24 pairs of consecutive Fibonacci numbers from above, compute the decimal number that results when the larger is divided by the smaller. Round your results to five decimal places.
6. The decimal numbers should be approaching a common value, the golden ratio. Find an approximation of the golden ratio correct to twenty-five digits.
7. For each member of your group, carefully measure the length of the longest bone of your middle finger (from knuckle to knuckle) and the length of the middle bone of your middle finger. Divide the larger number by the smaller number and record your results. Compare these values to the golden ratio.
8. Find some instances in nature where the golden ratio arises.