No
38
- Fibonacci's
Brothers
-
- We all know and love
the Fibonacci Numbers:
- 1 1 2 3 5 8 13 21 34
55 ...
- where each term is the
sum of the preceding two terms (zeros precede the initial
1)
- Some of you may even
know the wonders of the ratio of consecutive Fibonacci
Numbers:
|
1/1
|
1.00000...
|
|
2/1
|
2.00000...
|
|
3/2
|
1.50000...
|
|
5/3
|
0.66666...
|
|
8/5
|
1.60000...
|
|
13/8
|
1.62500...
|
|
21/13
|
1.61538...
|
|
34/21
|
1.61904...
|
|
55/34
|
1.61764...
|
|
89/55
|
1.61818...
|
|
144/89
|
1.61797...
|
|
233/144
|
1.61805...
|
|
377/233
|
1.61802...
|
- and it certainly
appears that these ratios are converging towards a particular
number.
- This number is called
The
Golden Ratio and is the
positive root of the quadratic equation:
- x2-x-1=0
- Check that this
appears to be true by calculating the root.
- It is also a root of
the cubic equation:
- x3-2x2+1=0
Generalising these
ideas produces the amazing Tribonacci Numbers:
- 1 1 2 4 7 13 24 44 81
149 ...
- where each term is the
sum of the preceding three terms (again with initial
zeros).
- The ratios of these
numbers approach 1.83929.... which is the root of the cubic
equation:
- x4-2x3+1=0
- and then there are the
Tetranacci Numbers ...
- ... the ratios
approach the root of the equation ...
- Do the calculations
and be amazed!
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do with this Number Fact then...
- E-mail us at madrascollege.maths@fife.gov.uk
- ...and we will publish them here.
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