Okay, here are the steps to find the 16th term: 1) 12th term is given as 144 2) 10th term is given as 55 3) 11th term is 144 - 55 = 89 4) 14th term is 89 + 144 = 233 5) 15th term is 144 + 233 = 377 6) 16th term is 233 + 377 = 610 So, the 16th term of the Fibonacci sequence is 610.

1. The Nature of
Mathematics
Mathematics in the Modern World

2. Activity!

3. Euclid is the
Father of
Mathematics

4. Euclid is the
Father of
Mathematics
Archimedes is regarded as
one of the most notable
Greek mathematicians. He
is known as the Father of
Mathematics.

5. If two expressions are equal to
each other, and you add the
same value to both sides of the
equation. Will the equation
would be remain equal ?

6. If two expressions are equal to
each other, and you add the
same value to both sides of the
equation. Will the equation
would be remain equal ?
Addition Property of Equality- If
two expressions are equal to
each other, and you add the
same value to both sides of
the equation, the equation will
remain equal

7. All quadrilaterals
are parallelograms

8. All quadrilaterals
are parallelograms
There are six basic types of
quadrilaterals. They are:
1.Trapezium
2.Parallelogram
3.Rectangle
4.Rhombus
5.Square
6.Kite

9. A point sometimes occupies
an area.

10. A point sometimes occupies
an area.
A point always
occupies an
area.

11. You can see Mathematics
everywhere!

12. Mathematics in our World
Mathematics
is …
a set of
problem-
solving
tools
a
language
an art
a study
of
patterns
a process
of
thinking
“Mathematics is the alphabet
with which God has written
the universe.”
- Galileo Galilei

13. The Elephant and the blind
men by James Baldwin
There were once six blind men who stood by the
road-side every day, and begged from the people who
passed. They had often heard of elephants, but they
had never seen one; for, being blind, how could they?
It so happened one morning that an elephant was
driven down the road where they stood. When they
were told that the great beast was before them, they
asked the driver to let him stop so that they might see
him.

14. The Elephant and the blind
men by James Baldwin
Of course they could not see him with their eyes; but
they thought that by touching him they could learn
just what kind of animal he was.
The first one happened to put his hand on the
elephant's side. "Well, well!" he said, "now I know all
about this beast. He is exactly like a wall."
The second felt only of the elephant's tusk. "My
brother," he said, "you are mistaken. He is not at all
like a wall. He is round and smooth and sharp. He is
more like a spear than anything else."

15. The Elephant and the blind
men by James Baldwin
The third happened to take hold of the elephant's
trunk. "Both of you are wrong," he said. "Anybody
who knows anything can see that this elephant is like
a snake."
The fourth reached out his arms, and grasped one of
the elephant's legs. "Oh, how blind you are!" he said.
"It is very plain to me that he is round and tall like a
tree."
The fifth was a very tall man, and he chanced to take
hold of the elephant's ear. "The blindest man ought
to know that this beast is not like any of the things
that you name," he said. "He is exactly like a huge
fan."

16. The Elephant and the blind
men by James Baldwin
The sixth was very blind indeed, and it was some time
before he could find the elephant at all. At last he
seized the animal's tail. "O foolish fellows!" he cried.
"You surely have lost your senses. This elephant is not
like a wall, or a spear, or a snake, or a tree; neither is
he like a fan. But any man with a par-ti-cle of sense
can see that he is exactly like a rope."
Then the elephant moved on, and the six blind men
sat by the roadside all day, and quarreled about him.
Each believed that he knew just how the animal
looked; and each called the others hard names
because they did not agree with him. People who
have eyes sometimes act as foolishly.

17. The Elephant and the blind
men by James Baldwin
To work mathematically, you need to smell the whole
elephant, hear its roar - and take pleasure in its
beauty, strength and also its surprising grace
and subtlety. And if we don't want to scare the
children? Well, who can resist a baby elephant?

18. Mathematics in our World
M a t h e m a t i c s i s a l l a r o u n d u s , i n
e v e r y t h i n g w e d o . I t i s t h e
b u i l d i n g b l o c k f o r e v e r y t h i n g i n
o u r d a i l y l i v e s , i n c l u d i n g m o b i l e
d e v i c e s , a r c h i t e c t u r e ( a n c i e n t
a n d m o d e r n ) , a r t , m o n e y ,
e n g i n e e r i n g , a n d e v e n s p o r t s .
Mathematics is the study of
assumptions, its properties and
applications.
Mathematics can also be the
science that deals with the
logic of shape, quantity and
arrangement.

19. 15 Incredible Examples of Mathematics in Nature
1. Snowflakes
The tiny but miraculous
snowflake, as an example of
symmetry in nature, exhibits six-
fold radial symmetry, with
elaborate, identical patterns on
each arm.
Snowflakes form because
water molecules naturally
arrange when they solidify. It’s
complicated but, basically, when
they crystallize, water
molecules form weak hydrogen
bonds with each other.
1. Sunflowers
Bright, bold and beloved by
bees, sunflowers boast
radial symmetry and a type
of numerical symmetry
known as the Fibonacci
sequence, 1, 2, 3, 5, 8, 13,
21, 24, 55, and so forth.
Scientists and flower
enthusiasts who have taken
the time to count the seed
spirals in a sunflower have
determined that the amount
of spirals adds up to a
Fibonacci number.

20. 15 Incredible Examples of Mathematics in Nature
1. Uteruses
According to a gynaecologist,
doctors can tell whether a uterus
looks normal and healthy based on
its relative dimensions – dimensions
that approximate the golden ratio.
When women are at their most
fertile, the ratio of uterus length
to its width is 1.6. This is a very
good approximation of the golden
ratio
1. Nautilus Shell
A nautilus is a cephalopod
mollusk with a spiral shell
and numerous short tentacles
around its mouth. Although
more common in plants, some
animals, like the nautilus,
showcase Fibonacci numbers.
A nautilus shell is grown in a
Fibonacci spiral. The spiral
occurs as the shell grows
outwards and tries to
maintain its proportional
shape.

21. 15 Incredible Examples of Mathematics in Nature
1. Romanesco Brocolli
Romanesco broccoli has an
unusual appearance, and
many assume it’s another
food that’s fallen victim to
genetic modification. However,
it’s actually one of many
instances of fractal
symmetry in nature. In
geometric terms, fractals
are complex patterns where
each individual component has
the same pattern as the
whole object. In the case of
romanseco broccoli, the entire
veggie is one big spiral
composed of smaller, cone-
like mini-spirals.
1. Pinecones
Pinecones have seed pods that
arrange in a spiral pattern.
They consist of a pair of
spirals, each one twisting
upwards in opposing directions.
The number of steps will
almost always match a pair of
consecutive Fibonacci numbers.
For example, a three–to–five
cone meets at the back after
three steps along the left
spiral and five steps along the
right. This spiralling Fibonacci
pattern also occurs in
pineapples and artichokes.

22. 15 Incredible Examples of Mathematics in Nature
1. Honeycombs
Honeycombs are an example of
wallpaper symmetry. This is
where a pattern is repeated
until it covers a plane. Other
examples include mosaics and
tiled floors.
Mathematicians believe bees
build these hexagonal
constructions because it is the
shape most efficient for storing
the largest possible amount of
honey while using the least
amount of wax. Shapes like
circles would leave gaps
between the cells because they
don’t fit perfectly together.
1. Tree Branches
The Fibonacci sequence is so
widespread in nature that it
can also be seen in the way
tree branches form and split.
The main trunk of a tree will
grow until it produces a branch,
which creates two growth
points. One of the new stems
will then branch into two,
while the other lies dormant.
This branching pattern repeats
for each of the new stems.

23. 15 Incredible Examples of Mathematics in Nature
1. Milky Way Galaxy
Symmetry and mathematical
patterns seem to exist
everywhere on Earth – the
Milky Way Galaxy was
discovered, and, by studying this,
astronomers now believe the
galaxy is a near-perfect mirror
image of itself.
Having mirror symmetry, the
Milky Way has another amazing
design. Like nautilus shells and
sunflowers, each ‘arm’ of the
galaxy symbolises a logarithmic
spiral that begins at the
galaxy’s centre and expands
outwards.
1. Faces
Humans possess bilateral symmetry.
Faces, both human and otherwise, are rife with examples of
the Golden Ratio. Mouths and noses are positioned at golden
sections of the distance between the eyes and the bottom
of the chin. Comparable proportions can be seen from the
side, and even the eye and ear itself, which follows along a
spiral. For example, the most beautiful smiles are those in
which central incisors are 1.618 wider than the lateral
incisors, which are 1.618 wider than canines, and so on.

24. 15 Incredible Examples of Mathematics in Nature
1. Orb Web Spiders
Orb web spiders create
near-perfect circular webs
that have near-equal-
distanced radial supports
coming out of the middle and
a spiral that is woven to
catch prey. Orb webs are
built for strength, with
radial symmetry helping to
evenly distribute the force
of impact when a spider’s
prey makes contact with the
web. This would mean there’d
be less rips in the thread.
1. Crop Circles
Crop circles are a sight to
behold because they’re so
geometrically impressive.
A study conducted by
physicist Richard Taylor
revealed that, somewhere in
the world, a new crop circle
is created every night, and
that most designs
demonstrate a wide variety
of symmetry and
mathematical patterns,
including Fibonacci spirals and
fractals.

25. 15 Incredible Examples of Mathematics in Nature
1. Starfish
Starfish or sea stars belong to
a phylum of marine creatures
called echinoderm. Other
notable echinoderm include sea
urchins, brittle stars, sea
cucumbers and sand dollars.
The larvae of echinoderms have
bilateral symmetry, meaning
the organism’s left and ride
side form a mirror image.
Sea stars or starfish are
invertebrates that typically
have five or more ‘arms’. These
radiate from an indistinct disk
and form something known as
pentaradial symmetry.
1. Peacocks
The peacock takes the
earlier principle of using
symmetry to attract a
mate to the nth degree.
Male peacocks utilise their
variety of adaptations to
seduce sultry peahens.
These include bright colours,
a large size, a symmetrical
body shape and repeated
patterns in their feathers.

26. 15 Incredible Examples of Mathematics in Nature
1. Starfish
Starfish or sea stars belong to
a phylum of marine creatures
called echinoderm. Other
notable echinoderm include sea
urchins, brittle stars, sea
cucumbers and sand dollars.
The larvae of echinoderms have
bilateral symmetry, meaning
the organism’s left and ride
side form a mirror image.
Sea stars or starfish are
invertebrates that typically
have five or more ‘arms’. These
radiate from an indistinct disk
and form something known as
pentaradial symmetry.
1. Peacocks
The peacock takes the
earlier principle of using
symmetry to attract a
mate to the nth degree.
Male peacocks utilise their
variety of adaptations to
seduce sultry peahens.
These include bright colours,
a large size, a symmetrical
body shape and repeated
patterns in their feathers.

27. 15 Incredible Examples of Mathematics in Nature
1. Sun-Moon Symmetry
The sun has a diameter of 1.4
million kilometres, while his
sister, the Moon, has a meagre
diameter of 3,474 kilometres.
With these figures, it seems
near impossible that the moon
can block the sun’s light and
give us around five solar
eclipses every two years.

28. FIBONACCI SEQUENCE AND
THE GOLDEN RATIO

29. Fibonacci, also called Leonardo Pisano,
English Leonardo of Pisa, original name Leonardo
Fibonacci, (born c. 1170, Pisa?—died after
1240), medieval Italian mathematician who
wrote Liber abaci (1202; “Book of the Abacus”),
the first European work on Indian and
Arabian mathematics, which introduced Hindu-
Arabic numerals to Europe. His name is mainly
known because of the Fibonacci sequence.
HISTORY

30. Fibonacci Sequence, (1,1,2,3,5,8,13,21, 34…) each number from the
sequence is the sum of the two numbers that precede it. Is strongly related
to the Golden Ratio ( = 1.618). First introduced by an Italian Mathematician
Leonardo Pisano Bigollo also known as Fibonacci in the year 1202.
Fibonacci observed numbers in nature. His most popular contribution
perhaps is the number that is seen in the petals of flowers.
Fibonacci Numbers

31. Fibonacci Sequence is an integer in the infinite sequence
1,1,2,3,5,8,13….. of which the first two terms are 1 and 1 and
each succeeding term is the sum of the two immediately
preceding
RULE: Add the first two term to get the next term
Fibonacci Sequence

32. •The Rabbit
So how many pairs
will be there in one
year?
1,1,2,3,5,8,13,21,34
,55,89,144,….,
So 144 is the 12th term, which
means that 144 pairs will be
there at the end of one year.

33. How is the Fibonacci sequence
derived?
1. First 2 terms (𝐹1 & 𝐹2) of the sequence is 1 and 1.
2. Add the last two consecutive terms to get the next term. Fibonacci sequence may also be
presented In squares starting with (1) and (1) meaning we have 1 pair of squares in which the
length of the side are all equal to one, the next square we have has a side of length (2) the next
square of course has side length of (3) followed by a square which has a length side of (5) a
square side length of 8,13,21,34 and so on, …

34. Fibonacci Sequence
𝐹𝑛−2 + 𝐹𝑛−1 = 𝐹𝑛, where n ≥ 3
𝐹10−2 + 𝐹10−1= 𝐹10
𝐹8 + 𝐹9 = 𝐹10
21 + 34 = 55
1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…

35. Find the indicated term of the
Fibonacci Sequence
❖9th term : 1,1,2,3,5,8,13,21,___,
❖15th term :
❖21st term:
34
610
10946

36. Determine the 16th term of the
Fibonacci number given that the 12th
term is 144 and the 10th term is 55.
𝐹𝑛−2 + 𝐹𝑛−1 = 𝐹𝑛
𝐹12 = 𝐹11 + 𝐹10
144 = 𝐹11 + 55
144 − 55 = 𝐹11
𝐹11 = 89

37. Determine the 16th term of the
Fibonacci number given that the 15th
term is 610 and the 17th term is 1597.
𝐹𝑛−2 + 𝐹𝑛−1 = 𝐹𝑛
𝐹17 = 𝐹16 + 𝐹15
1597 = 𝐹16 + 610
1597 − 610 = 𝐹11
𝐹17 = 987

38. Fibonacci Sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, 233, 377, 610, 987, 1597, 2584,
4181, 6765, 10946, 17711, 28657,
46368, 75025, 121393, 196418,, ...
What is the number
after 196418?

39. Fibonacci Sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, 233, 377, 610, 987, 1597, 2584,
4181, 6765, 10946, 17711, 28657,
46368, 75025, 121393, 196418,
317811, ...What is the number
after 196418?

40. Since that Fibonacci
sequence is an infinite
sequence, how can we find
the nth Fibonacci number

41. 𝐹𝑛 =
1 + 5
2
𝑛
−
1 − 5
2
𝑛
5
To find the nth term of the Fibonacci
Sequence use the Binet’s Formula

42. 𝐹𝑛 =
1 + 5
2
𝑛
−
1 − 5
2
𝑛
5
Find the 50th term of the Fibonacci
numbers
𝐹50 =
1 + 5
2
50
−
1 − 5
2
50
5
≈12586269025.00002 or 12,586,269,025

43. 𝐹𝑛 =
1 + 5
2
𝑛
−
1 − 5
2
𝑛
5
Find the 100th term of the Fibonacci
numbers
𝐹100 =
1 + 5
2
100
−
1 − 5
2
100
5
354,224,848,179,261,915,075

44. 𝐹𝑛 =
1 + 5
2
𝑛
−
1 − 5
2
𝑛
5
Find the 80th term of the Fibonacci
numbers
𝐹50 =
1 + 5
2
80
−
1 − 5
2
80
5
14,472,334,024,676,221

45. THE GOLDEN RATIO
•In mathematics and the arts, two quantities are in the golden ratio if the
ratio between the sum of those quantities and the larger one is the same
as the ratio between the larger one and the smaller.
• In this case, we refer to a very important number that is
known as the golden ratio.
• The golden ratio is a mathematical constant
approximately 1.6180339887.

46. THE GOLDEN RATIO
It is interesting to note that the ratio of
two adjacent Fibonacci numbers
approaches the golden ratio; that is,
𝐹𝑛
𝐹𝑛−1

47. THE GOLDEN RATIO
Putting it as simply as we can (eek!), the
Golden Ratio (also known as the Golden
Section, Golden Mean, Divine Proportion
or Greek letter Phi) exists when a line is
divided into two parts and the longer part
(a) divided by the smaller part (b) is equal
to the sum of (a) + (b) divided by (a), which
both equal 1.618.

48. THE GOLDEN RATIO

49. THE GOLDEN RATIO

50. THE GOLDEN RATIO
Application
Do you want to have a perfect collage of photos that will be
perfect on your social media account? Use the Golden Ratio
diagram.

51. Thank you !!
“Mathematics is the alphabet with which God has
written the universe”
-Galileo Galilei
“Mathematics is the alphabet
with which God has written
the universe.”
- Galileo Galilei