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Broad Topics > Sequences, Functions and Graphs > Fibonacci sequence

1 Step 2 Step

Stage: 3 Challenge Level:

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?

Fibs

Stage: 3 Challenge Level:

The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

Fibonacci Surprises

Stage: 3 Challenge Level:

Play around with the Fibonacci sequence and discover some surprising results!

Farey Fibonacci

Stage: 5 Short Challenge Level:

Investigate Farey sequences of ratios of Fibonacci numbers.

Cellular

Stage: 2, 3 and 4 Challenge Level:

Cellular is an animation that helps you make geometric sequences composed of square cells.

Building Gnomons

Stage: 4 Challenge Level:

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

Simple Train Journeys

Stage: 1 and 2 Challenge Level:

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

Golden Trail 1

Stage: 3, 4 and 5 Challenge Level:

A first trail through the mysterious world of the Golden Section.

First Forward Into Logo 11: Sequences

Stage: 3, 4 and 5 Challenge Level:

This part introduces the use of Logo for number work. Learn how to use Logo to generate sequences of numbers.

Whirling Fibonacci Squares

Stage: 3 and 4

Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected.

Plus or Minus

Stage: 5 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

Pythagorean Fibs

Stage: 5 Challenge Level:

What have Fibonacci numbers got to do with Pythagorean triples?

Fibonacci Fashion

Stage: 5 Challenge Level:

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

Golden Mathematics

Stage: 5

A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.

The Golden Ratio, Fibonacci Numbers and Continued Fractions.

Stage: 4

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

Golden Fractions

Stage: 5 Challenge Level:

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

Sheep Talk

Stage: 2 Challenge Level:

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?

Got a Strategy for Last Biscuit?

Stage: 3 and 4 Challenge Level:

Can you beat the computer in the challenging strategy game?

Leonardo of Pisa and the Golden Rectangle

Stage: 2, 3 and 4

Leonardo who?! Well, Leonardo is better known as Fibonacci and this article will tell you some of fascinating things about his famous sequence.

Fibonacci's Three Wishes 2

Stage: 2 and 3

Second of two articles about Fibonacci, written for students.

Golden Fibs

Stage: 5 Challenge Level:

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

Fibonacci Factors

Stage: 5 Challenge Level:

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

Ordered Sums

Stage: 4 Challenge Level:

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. . . .

Stringing it Out

Stage: 4 Challenge Level:

Explore the transformations and comment on what you find.

Stage: 3 and 4 Challenge Level:

Using logo to investigate spirals

LOGO Challenge - Circles as Bugs

Stage: 3 and 4 Challenge Level:

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

Continued Fractions I

Stage: 4 and 5

An article introducing continued fractions with some simple puzzles for the reader.

Gnomon Dimensions

Stage: 4 Challenge Level:

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

Golden Powers

Stage: 5 Challenge Level:

You add 1 to the golden ratio to get its square. How do you find higher powers?

Room Doubling

Stage: 2 Challenge Level:

Investigate the different ways you could split up these rooms so that you have double the number.