v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} Normal 0 false false false false EN-US X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable {mso-style-name:”Table Normal”; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:””; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:”Arial”,sans-serif; mso-ansi-language:EN-GB;} A: For each equation:

1. Solve for  in terms of .                                                                                         (4)
2. Find the value of the ratio                                                                                       (4)
3. Write the ratio  in decimal form, correct to three decimal places where possible.  (4)

B: For each pair of simultaneous equations:

and

and

1. Solve for  and                                                                                                    (3)
2. Find the ratio .                                                                                                      (4)
3. Write the ratio  in decimal form correct to three decimal places where possible. (1)

You may have noticed that the numbers 1, 2, 5, 8 and 13 have appeared as solutions in Task 1. These numbers form a sequence called the Fibonacci sequence.  In the Fibonacci sequence each number after the second term is the sum of the two before it. The table shows the first eight terms of the sequence (T1=term 1)

T1

T2

T3

T4

T5

T6

T7

T8

1

1

2

3

5

8

13

21

1. Write down the next three terms of the sequence.                                                (3)
2. Find a new sequence by finding the ratio.

up to                                                                                        (2)

1. Find the ratios in Question 2 correct to three decimal places.                               (2)
2. Describe what you notice?                                                                                     (1)

We can generate a sequence of numbers by continued square roots:

1. Use a calculator (or Excel) to find the first ten terms of this sequence for .    Give your answers correct to four decimal places.

Calculator tip:

For T1 press ; for T2 press ; for T3 press  and so on, for each successive term.

Write down each term’s value in a table.                                                                   (2)

1. Find the first 11 terms of the same sequence for  and .

Record these values in the same table.                                                                     (4)

1. Describe what you notice about the values of the terms.                                          (2)

We have seen the number 1,618 (1,61803398 to be more accurate), in three different situations

●     the ratio of the terms of the Fibonacci sequence

●     continued square roots

●     continued fractions.

This number is called ‘phi’ or .  It is an important number and is also called the Golden ratio.  has many special properties.  Let

1. Write down the value of                                                                            (1)
2. What do you notice?                                                                                           (2)
3. Use the fact that  to solve for  and show that .          (3)

The Golden rectangle is a rectangle in which the ratio of the sides is  of approximately 1,618.

Throughout the ages this ratio has been recognised as aesthetically pleasing.  Many buildings and works of art have the Golden rectangle in them.  Two examples are the Parthenon in Greece and the United Nations building in New York, where the ratio of the width of the buildings compared with the height of every ten floors is the Golden ratio.

Rectangle 1 has sides  and  with the ratio of length to breadth of

Rectangle 2 is formed by creating a square of length  on the one side of Rectangle 1.  The remaining Rectangle 2 has sides of length and  with the ratio of length to breadth of ..  If the two rectangles have the same ratio of length to breadth, then they are Golden rectangles.  In other words .

1. Show that the equation is the same as                                      (2)
2. If , solve for  in the equation. Leave your answer in surd form.

(Why can we discard one of the values of ?)                                                         (3)