THE ART AND SCIENCE OF MATHEMATICS
Wednesday, September 28, 2005

The Fibonacci Sequence

As we all know, the Fibonacci numbers $\{F_n\}$ satisfy the recurrence relation

$$a_{n+2} = a_{n+1} + a_n ,$$ (1)

with $F_1=1$ and $F_2=1$. The Fibonacci numbers are not the only solutions to this recurrence relation; different choices for $a_1$ and $a_2$ will give different sequences.

Is there is a solution to this recurrence relation of the form

$$a_n = \lambda^{n-1}$$ (2)

for $n=1,2,\ldots$ Since we take $a_1=\lambda^0$, which we define to be one even if $\lambda=0$, the trivial solution $a_n=0^{n-1}$ does not solve equation (1).

Problems

1. Find the nontrivial solutions $\lambda=\lambda_1$ and $\lambda=\lambda_2$ so that $a_n=\lambda^{n-1}$ satisfies equation (1).
2. Show that $a_n=c \lambda^{n-1}$ solves (1) for any constant $c$, provided $\lambda=\lambda_1$ or $\lambda=\lambda_2$.
3. Show that $a_n=c_1 \lambda_1^{n-1} + c_2 \lambda_2^{n-1}$ solves (1) for any constants $c_1$ and $c_2$.
4. Can you find constants $c_1$ and $c_2$ such that the Fibonacci numbers $\{F_n\}$ are given by
   

   $$F_n = c_1 \lambda_1^{n-1} + c_2 \lambda_2^{n-1}$$ (3)

   
   
   for every $n$?
5. Consider a rectangle with sides of length $x$ and $y$, where $x<y$ and $\frac{y}{x}$ is equal to $\Phi$. Remove a square with sides of length $x$ from the rectangle. Show that the ratio of the sides of the remaining rectangle is also equal to the golden ratio.

Further reading

• The powerpoint presentation can be found at

  http://eaton.math.rpi.edu/CourseMaterials/spring02/artscience/

• Another website with a wealth of material:

  http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html