The Fibonacci
Sequence and the
Golden Ratio
math is all around you.
Can we find a pattern?
1, 1, 2, 3, 5, …
8, 13, 21, 34, 55, …
What comes next?
What is the pattern?
The Fibonacci Sequence:
1, 1, 2, 3, 5, 8, 13, 21, …
How would you describe the pattern in
words?
Next = Current + Previous
𝑎𝑛+1 = 𝑎𝑛 + 𝑎𝑛−1
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
1
1
=1;
2
1
=2;
3
2
= 1.5
5
= 1. 66
3
21
= 1. 615384
13
8
=
5
34
= 1. 619047
21
1.6
13
= 1.625
8
55
= 1.61764706
34
Here are the next few:
89
= 1.6181818
55
144
= 1.6179775
89
233
= 1.6180555
144
The Golden Ratio
• They are getting closer to the same
number:
1+ 5
𝜑=
≈ 1.61803398875
2
• We call this number the “Golden Ratio”
• Why do you think we use the word ratio?
Why is the Golden Ratio Important?
Why do Plants do This?
http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html
https://www.youtube.com/watch?v=ahXIMUkSXX0
Now for a change of Pace
• Remember the quadratic formula?
2
• If 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 then 𝑥 =
−𝑏± 𝑏2 −4𝑎𝑐
2𝑎
• Let’s use it on the equation:
𝑥2 − 𝑥 − 1 = 0
• Solution:
1± 5
𝑥=
2
Why is that?
• From Earlier: 𝑎𝑛+1 = 𝑎𝑛 + 𝑎𝑛−1
𝑎𝑛+1 𝑎𝑛 + 𝑎𝑛−1
=
𝑎𝑛
𝑎𝑛
𝑎𝑛+1 𝑎𝑛 𝑎𝑛−1
=
+
𝑎𝑛
𝑎𝑛
𝑎𝑛
𝑎𝑛+1
𝑎𝑛−1
=1+
𝑎𝑛
𝑎𝑛
𝑎𝑛+1
1
=1+ 𝑎
𝑛
𝑎𝑛
𝑎𝑛−1
𝑎𝑛+1
1
=1+ 𝑎
𝑛
𝑎𝑛
𝑎𝑛−1
1
φ=1+
φ
1
φ (φ = 1 + )
φ
φ2 = φ + 1
φ2 − φ − 1 = 0