A Fibonacci pattern

Recall the Fibonacci numbers,

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \dots,

the sequence of numbers beginning with 0, 1 where each subsequent number is the sum of the previous two:

\begin{array}{rcl}F_0 & = & 0 \\ F_1 & = & 1 \\ F_n & = & F_{n-1} + F_{n-2} \qquad (n \geq 2). \end{array}

Try this: pick any Fibonacci number. Square it. Now, look at the two Fibonacci numbers on either side of your chosen one, and multiply them. For example, suppose we pick 13. 13^2 = 169. The two Fibonacci numbers on either side of 13 are 8 and 21; their product is 8 \cdot 21 = 168.

Try this with a few different Fibonacci numbers. Better yet, make a table of your results for the first n Fibonacci numbers. What do you notice? Can you express your observation algebraically? Can you prove it is true for every Fibonacci number? What does this have to do with the area paradox?

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4 Responses to A Fibonacci pattern

  1. Classic, great stuff. Have used this with teacher education students to good effect. Am hoping to get h.s. teachers I coach in Detroit to use it. Thanks for putting it up.

  2. Pingback: Cassini’s identity | The Math Less Traveled

  3. xander says:

    As it is the end of the semester, I’ve been a little busy, but I turned in grades on Friday, and finally got a chance to spend some time with this problem. Rather than clutter up your comments with junk, I’ll pimp my own website: my response to your original set of questions.

    One other question: what plugin are you using for LaTeX input?

    • Brent says:

      Thanks, nice commentary! Since this blog is hosted on wordpress.com I use whatever LaTeX plugin they provide by default, I am not sure which it is.

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