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.

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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.

12. SEQUENCE IN NATURE

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
𝑎 𝑛 =
1
𝑎1 + (𝑛 − 1)𝑑
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.

15. HARMONIC SEQUENCE
.