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:
B: For each pair of simultaneous equations:
and
and
TASK 2 [8]
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
up to (2)
TASK 3 [8]
We can generate a sequence of numbers by continued square roots:
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)
Record these values in the same table. (4)
TASK 4 [6]
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
TASK 5 [8]
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 .
(Why can we discard one of the values of ?) (3)
Use the equation to show that , use golden ratio
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