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Connecting Fibonacci and Pythagoras


I assume you know about the Fibonacci Numbers:

1,1,2,3,5,8,13,21,34,55,89,...
where each term is the sum of the two preceding terms.

Consider this sequence of four consecutive Fibonacci numbers: 3, 5, 8, 13.

Mulitply the outside two numbers: 3 x 13 = 39

Double the product of the inside two numbers: 2 x 5 x 8 = 80

Now, let 39 and 80 be the lengths of the two shorter sides of a right triangle...what is its hypotenuse? 89...another Fibonacci number!

What is the area of the triangle....1560 = 3x5x8x13...is it a Fibonacci number?

First, why isn't the area result surprising at all?

Second, choose another sequence of four consecutive Fibonacci numbers. Will this process again produce a hypotenuse that is a Fibonacci number as well? Explain.

Third, will the process work for some/all sequences of four consecutive Fibonacci numbers? Explain...prove?

And fourth, suppose you had a more general version of the Fibonacci sequence, where you start with any two generators you want...e.g. 3,7,10,17,27,... Does the above process still work? Explain.

 


Hint: Try more sequences of four consecutive Fibonacci numbers...look for an interesting pattern that "should" emerge?

 


Solution Commentary: In the first example, the sides are 3x13 = 39, 2x5x8 = 80, and 89...now, did you discover that 89 = (3x8)+(5)(13)....the sum of the alternating products of the four original numbers. Now, does this always happen?

Suppose the sequence was 13, 21, 34, 55...The two sides are 13x55 and 2x21x34...is the resulting hypotenuse the sum (13x34)+(21x55)?

This should give you a great start...