6.4 IN-DEPTH PATH calculations with letters
Theory
Like terms
Like terms are "things" which are alike (the same).
Pears are the same --> They are like terms
The letter P is the same --> They are like terms
There are a couple of rules:
* You may only add and subtract like terms.
* You may always multiply.
* 3 . 3 = 32 , x . x = x2 , a . a = a2
Test yourself exercise
You really need to understand this theory in order to understand the new theory about the Fibonacci sequence. Therefore, you are going to test yourself with this exercise.
Simplify
5ab + 6a + 7b + 3ab + 2a . -3b + 4a . 8a
Fibonacci sequence & Lucas sequence
Extra learning objectives
At the end of this paragraph you are able to...
...explain what the Fibonacci sequence is.
...explain what the Lucas sequence is.
...calculate with the Fibonacci sequence.
...calculate with the Lucas sequence.
...use letters instead of numbers with the Fibonacci sequence.
...explain what the golden ratio is.
...explain where you can see the golden ratio.
The Fibonacci sequence
The Italian mathematician Leonardo of Pisa, now known as Fibonacci, was the first to introduce the Fibonacci sequence.
The Fibonacci sequence appeared when he was finding a solution for a problem:
" A man puts a pair of rabbits in a place surrounded by walls. They wanted to know how many pairs of rabbits could be produced from that certain pair in a year. It was supposed that every month each pair would become a new pair, which after two months would become productive."
Fibonacci found the resulting sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... as answer. He could explain his sequence by the picture underneath.
A picture of the rabbit problem and its solution is shown above. After month 5 the sequence continues.
The Fibonacci sequence starts with a 0, 1 and you need to add the latest two to get the next one.
Now have a look at the pattern of the Fibonacci sequence.
The Lucas sequence
The Lucas sequence is a sequence named after the mathematician François Édouard Anatole Lucas. Lucas was studying the Fibonacci sequence and that is when he found a similar series occur:
2, 1, 3, 4, 7, 11, 18, ...
Instead of beginning with 0,1 Lucas starts with 2,1. The pattern is still the same as the Fibonacci sequence, you need to add the latest two terms to get the next one.
The golden ratio
There is a link between the Fibonacci numbers and the Golden ratio.
The Fibonacci numbers:
0 + 1 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
13 + 21 = 34
21 + 34 = 55
34 + 55 = 89
55 + 89 = 144
89 + 144 = 233
etc...
You can calculate the ratio by dividing the two immediately previous terms.
Examples:
8 : 5 = 1.6
13 : 8 ≈ 1.6
21 : 13 ≈ 1.6
34 : 21 ≈ 1.6
21 : 34 ≈ 1.6
34 : 55 ≈ 1.6
etc...
The answers are each time approximately 1.6. Actually, the number is each time 1.61803398875..... and it is called phi. Phi is a never ending number and that's why we call it phi instead of 1.618033.....
Phi is the golden ratio or the golden number.
This ratio could be expressed as a 1 : 1.6 (a 1 to 1.6 ratio) and is called the Golden ratio.
In the picture above, a 1:1 ratio is shown compared to a 1:1.6 ratio.
To get the 1 : 1.6 ratio you need to take a little more than half of the initial line added to the line on the other side.
You can see what the golden ratio looks like in a rectangle:
Within this rectangle you can make other rectangles with the golden ratio. If you continue to do this you will get a golden ratio spiral.
The golden ratio has a lot to do with everyday life! You can see it in nature, music, architecture, art and even in the human body!
Look at the pictures below and try to find the golden ratio or the golden ratio spiral.
Video golden ratio
Look at the video to find out more about the golden ratio.
The Lucas sequence with letters
You have already learned how the Lucas sequence looks like:
2, 1, 3, 4, 7, 11, ...
We can also use numbers instead of letters:
a, b, a + b, a + 2b, 2a + 3b, 3a + 5b, ....
Exercises
Exercise 1:
Simplify
a) 10pq + pq . 43 - 2
b) 5gh . g . 4 . a
c) 6a + 7 . 2a - 5
Exercise 2:
Complete
a) 6a + ... = 8a
b) 9b . ... = 18b2
c) -1/3ab . ... = 18a2b
Exercise 3:
Several Lucas sequences are shown below.
Write down the next six numbers.
a) b, c, b + c, b + 2c, ...
b) k + s, 2k, 3k + s, 5k + s, ...
c) 6m + 2, 5q + 1, ...
d) h + j, h + j, ...
