Math project
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The document discusses Fibonacci sequences and their relationship to the golden ratio. It provides Binet's formula for calculating the nth term of the Fibonacci sequence using the golden ratio and its reciprocal. Examples are given to demonstrate calculating the number of choices after a given number of months using the Fibonacci sequence. The document also discusses harmonic sequences and their history relating to music, providing the general rule for calculating terms of a harmonic sequence.
Math project
- 1. 1,1,2,3,5,8,_,_,34,_ 1,1,2,3,5,8,13,21,34,551 3 , , , , 1 243 1 3 , 1 9 , 1 27 , 1 81 , 1 243 Find the missing terms What did you observe?
- 2. Jonas Amante Jerome Galve Lea Bernal Lara Escarcha Marjorie Nogales 10-Newton
- 3. FIBONACCI SEQUENCE - Each successive number is the addition of the previous two numbers in the sequence. ASSOCIATED WITH THE GOLDEN RATIO represented by the Greek letter phi Relation to the Fibonacci sequence Named after 13th Century Italian mathematician Leonardo Fibonacci
- 4. FIBONACCI SEQUENCE Binet's Formula for the nth Fibonacci number Involving golden section number Phi and its reciprocal phi:
- 5. FIBONACCI SEQUENCE Binet's Formula for the nth Fibonacci number Using just one of the golden section values: Phi, and all the powers are positive:
- 6. EXAMPLES A boat company wants to make 2 kinds of boat. First a canoe which takes a month to make and a sailing dinghy which takes 2 months to build. They only have enough space to build one boat at a time but it does have plenty of customers waiting. Suppose the area where the boats are built ahs to be closed for maintenance soon: Close after a month = 1 canoe Close after 2 months = 2 canoes/1 dinghy Close after 3 months = 3 canoes/ dinghy first then canoe/canoe first then dinghy The question is How many choices are there after 6 months?
- 7. SOLUTION Given: Close after a month = 1 canoe Close after 2 months = 2 canoes/1 dinghy Close after 3 months = 3 canoes/ dinghy first then canoe/canoe first then dinghy Required: Number of choices after 6 months
- 8. SOLUTION Number of months Number of choices 1 1 2 2 3 3 4 5 5 8 6 13 F(4)=F(3)+F(2) 5= 3+2 F(4) = 5 F(5) = F(4)+F(3) 5+3= 8 F(5)=8 F(6)=F(5)+F(4) F(6) = 13 There will be 13 choices after 6 months.
- 9. EXAMPLES In a hive, thousands of honeybees can be found. In a colony, there is a certain female called the queen which is called special. There are also workers but unlike the queen, they can't produce eggs. There are also some drone bees who are male but doesn't work. The males were produce by the queen only so it means that they have a mother but no father. The femals were also produce by the queen but it is when she mated with a male so it means that they have a mother and a father. Most of them ends up as workers but some are fed with a royal jelly which makes them grow and turn into queens. When that happens, they can start their own hive. Now, let's look at the family of a male drone bee: 1. He had 1 parent, a female. 2. He has 2 grand-parents, since his mother had two parents, male and female. 3. He has 3 great grand parents: his grand-mother had two parents but his grand- father had only one The question is "How many great-great-grand parents did he have?"
- 10. SOLUTION Given: 1. He had 1 parent, a female. 2. He has 2 grand-parents, since his mother had two parents, male and female. 3. He has 3 great grand parents: his grand-mother had two parents but his grand-father had only one Required: Number of great-great-grand parents he had Family of a male drone bee
- 11. SOLUTION Numbe r of Parents Grand- parents Great- grand- parents Great- Great- Grand- parents Male 1 2 3 5 Numbe r of Parents Grand- parent s Great- grand- parents Great- Great- Grand- parents Male 1 2 3 5 Female 2 3 5 8 Solution: F(4)=F(3)+F(2) 5= 3+2 F(4) = 5 He had 5 Great-Great-Grandparents which 3 less than a female had.
- 13. HARMONIC SEQUENCE A sequence of numbers a1, a2, a3, such that their reciprocals 1/a1, 1/a2, 1/a3, form an arithmetic sequence (numbers separated by a common difference). History: The study of harmonic sequences dates to at least the 6th century bce, when the Greek philosopher and mathematician Pythagoras and his followers sought to explain through numbers the nature of the universe. One of the areas in which numbers were applied by the Pythagoreans was the study of music. In particular, Archytas of Tarentum, in the 4th century bce, used the idea of regular numerical intervals to devise a theory of musical harmony (from the Greek harmonia, for agreement of sounds) and the enharmonic method of tuning musical instruments.
- 14. HARMONIC SEQUENCE General rule for Harmonic sequence = 1 1 + ( 1) Example : The sequence . The sum of a sequence is known as a series, and the harmonic series is an example of an infinite series that does not converge to any limit. That is, the partial sums obtained by adding the successive terms grow without limit, or, put another way, the sum tends to infinity.