Fibonacci sequence

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.

In sheep talk the only letters used are B and A. A sequence of words is formed by following certain rules. What do you notice when you count the letters in each word?

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

Explore the transformations and comment on what you find.

Using logo to investigate spirals

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.