THE ART AND SCIENCE OF MATHEMATICS October 12, 2011

The Fibonacci Sequence

As we all know, the Fibonacci numbers {Fn} satisfy the recurrence relation

an+2 = an+1 + an,
(1)

with F0 = 0, F1 = 1 and F2 = 1. The Fibonacci numbers are not the only solutions to this recurrence relation; different choices for a0 and a1 will give different sequences.

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

n an = λ
(2)

for n = 1, 2,…? (Since we take a0 = λ0, which we define to be one even if λ = 0, the trivial solution an = 0n does not solve equation (1).)

Problems

  1. Find the nontrivial solutions λ = λ1 and λ = λ2 so that an = λn satisfies equation (1).
  2. Show that an = cλn solves (1) for any constant c, provided λ = λ 1 or λ = λ2.
  3. Show that an = c1λ1n + c 2λ2n solves (1) for any constants c 1 and c2.
  4. Can you find constants c1 and c2 such that the Fibonacci numbers {Fn} are given by
    Fn = c1λn1 + c2λn2
    		(3)

    for every n?

  5. Consider a rectangle with sides of length x and y, where x < y and y x is equal to Φ. 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