The three rectangles in the picture are split into squares.

Assuming that the smallest sized square has sides of length 1, what is the ratio of the two sides of each of the three rectangles?

What is the length of each of the rectangle's sides if the smallest squares have sides of length 2?

3. In the continued fraction for 45/16 = [2; 1, 4, 3], let's shall see what happens when change the final 3 to another number. Can you spot the pattern?

Convert the following to proper fractions:

o |
[2 |
1, |
4, |
4] |

o |
[2 |
1, |
4, |
5] |

o |
[2 |
1, |
4, |
6] |

o |
[2 |
1, |
4, |
7] |

o |
[2 |
1, |
4, |
n] |

How is your pattern related to the proper fraction for [2; 1,4 ]?

4. Convert these pairs of continued fractions into a single proper fraction:

o [0; 1,2,3] and [0; 1,2,2,1] o [1; 1,2] and [1; 1,1,1] o [3; 2] and [3; 1,1] What is the general principle here?

5. Here is the Fibonacci Spiral from the Fibonacci Numbers in Nature page:

If the smallest squares have sides of length 1, what continued fraction does it correspond to? What proper fraction is this?

6. Convert the successive Fibonacci number ratios into continued fractions You should notice a striking similarity in your answers.

If the ratio of consecutive Fibonacci numbers gets closer and closer to Phi, what do you think the continued fraction might be for Phi=1-618034... which is what the above fractions are tending towards?

7. The last question made fractions from neighbouring Fibonacci numbers. Suppose we take next-but-one pairs for our fractions, eg

etc.

o Convert each of these to continued fractions, expressing them in the list form. What pattern do you notice?

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