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

Here are some circle bugs to try to replicate with some elegant programming, plus some sequences generated elegantly in LOGO.

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.