Fibonacci

THE FIBONACCI SEQUENCE About 800 years ago, there was a fellow named Leonardo Fibonacci who lived in a small Italian town named Pisa, world-renowned these days for its leaning tower and cheesy tomato pies. Leonardo was the only guy in town who knew how to add and multiply, so he became famous as a mathematician. A farmer, wishing to protect his zucchini crop from a possible rabbit invasion, came to Leonardo hoping he could solve this problem: A pair of rabbits are put in a field and, if rabbits take a month to mature and then produce a new pair every month after that, how many pairs will there be in twelve months time? Leonardo did not like people giving him problems. All he wanted to do was add and multiply with the new numbers he learned while a student in Africa. But after groaning a bit, he started thinking: At the end of the first month, the bunnies are not old enough to reproduce, so there is still only 1 pair of bunnies. At the end of the second month the female produces a new pair of bunnies, so now there are 2 pairs of bunnies running lose in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. If this keeps up for 12 months, the poor farmer is going to be up to his ears in bunnies. The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... So Leonardo made a list for the farmer like the one below showing how the bunny population grew over a 20 month period. Over the years, Leonardo found other interesting patterns hidden among the numbers in his original list, and made more lists. Fill in the columns below and reproduce Leonardo's lists. Use this table to answer the questions on this page. Find the pattern in the first column. Then complete each column in the table above. Copy the finished table. F(n)xF(n) means to multiply what is in the first column by itself. F(n)xF(n+1) means to multiply what is in the first column by the term just below it. F(n+1)/F(n) means to divide what is in the first column into the term just below it. The Golden Rectangle Click the button till you draw a rectangle. Use the table above to answer the questions below: QUESTIONS If AC is 1 inch long... How long is CD? How long is MD? How long is MC? (Hint: Use the Pythagorean Theorem: a² + b² = c² ) What is half the square root of 5? How long is ME? How long is BE? How long is 1 divided by BE? How long is DE? How long is 1 divided by DE? Is this number in the table above? Where? Using Google, find a building whose width divided by its height equals BE or DE? More Golden Rectangles Find the largest rectangle above by clicking inside the empty square. Notice that the rectangle is composed entirely of squares. The number in the center of a square is the width and length of the square. Use the table above to answer the questions below: What is the sum of all the square areas? What is the area of the next larger rectangle? Which column in the table above computes the area of each of the small squares? What is the ratio of the largest rectangle's long side to its short side? A Few Golden Squares Find the largest square above by clicking inside the empty rectangle. Notice that the square is composed entirely of rectangles. The number in the center of a rectangle is the width of the rectangle. Use the table above to answer the questions below: What is the sum of all the rectangular areas? What is the area of the next larger square? Which column in the table above computes the area of each of the small rectangles? What is the ratio of the largest rectangle's long side to its short side? Find this ratio in the table you made. Complete Pattern A in the table above. 20. Guess the sum of the first 10 Fibonacci terms without adding them. Complete Pattern B in the table above. 21. Write the next line of Pattern B. Complete the last line of Pattern C in the table above. 22. Guess the sum of the squares of the first 10 Fibonacci terms without adding them. Complete Pattern D in the table above. 23. Write the next line of Pattern D. 24. Which of the 4 patterns above, A B C D, best describes the construction of the rectangles in the mathlet above titled More Golden Rectangles? 25. What is the value of (1+√5)/2 ? This number is called the Golden Ratio and is designated by its own special symbol φ. Store this number in your calculator's memory. You will need it to complete the table above. 26. Complete the table above using this number φ in the formula φn / √5. Round off the first column of numbers to their nearest integer. Enter these integers in the last column. You should be able to guess by now what they are. But can you prove the formula is true for all n?

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From Lydesia "de" [198.146.87.95] - 9/30/08 12:59 PM