The document discusses the Fibonacci sequence and its properties. It begins by explaining how the Fibonacci sequence is defined, with each subsequent number being the sum of the previous two numbers. It then provides examples of calculating Fibonacci numbers. The document also discusses how the Fibonacci sequence appears in nature, such as the spiral patterns of sunflowers and pinecones. Finally, it notes that the ratio of adjacent Fibonacci numbers approaches the golden ratio, an interesting mathematical property.

1. StUdEnT = Anushka Sahu
StAnDaRd = Tenth ‘ A ’
EnRoLlMeNt = 10015
SuBjEcT = Mathematics
ToPiC = FIBONACCI SEQUENCE

3. QUESTIONAIRRE
1. Let's begin with a spot of math- it has been mathematically proven that the
recurring decimal 0.999... (with an infinite number of 9s) is *exactly* equal
to ___.
• 95
• 7 and a quarter
• 1
• 17
2. Several religions, such as Judaism and Islam, are said to be 'monotheistic'.
How many 'gods' does each of these believe in?
Answer: (a number)
3. All natural numbers (1, 2, 3, 4, 5...) can grouped as 'odd' and 'even'. 'Even
numbers' are specifically those numbers which, when divided by ___, give a
remainder of zero.
• 1
• 2
• 7
• 42009

4. 4. A 'duo' is a group of *two* people or things. For example, Batman and Robin
are sometimes called the 'Dynamic Duo'. But how many are there in a *trio*?
Answer: (a number)
5. The UN Security Council has 15 members in all. Out of these, how many are
*permanent* members, enjoying veto powers?
• -1.73
• 0
• 14
• 5
6. Complete the lyrics for the famous Christmas song, "The Twelve Days of
Christmas": "___ maids a-milking..."
• 8
• 10
• 6
• 9

5. 7. This number is considered 'unlucky' by most cultures, but is 'lucky' for traditional Chinese
people. At the Last Supper, this was the number of people (including Christ) who were present.
A 'baker's dozen' has this number of loaves. Which number am I talking about?
• 6
• 13
• 12
• -1
8. The popular card game 'Blackjack', known as 'Pontoon' in the UK, is also called ___.
• 17
• 70
• 3
• 21
9. The element Selenium (Se) has an electronic configuration of '2-8-18-6'. What is its atomic
number?
• Sir Humphrey Appleby
• 65,536
• 34
• 1
10. Now, if you've noticed the pattern which the answers in this quiz are forming, you should
be able to get this one!
What is the 7th root of 1522435234375?
• Answer: (a number)

6. ANSWERS
1) 1
2)1
3)2
These Answers
4)3
5)5 form a Fibonacci
6)8 Series !
7)13
8)21
9)34
10) 55

7. FIBONACCI SEQUENCE
• The Fibonacci Sequence is the series of numbers:
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, ...
1.The next number is found by adding up the two numbers before it.
2.The 2 is found by adding the two numbers before it (1+1)
3.Similarly, the 3 is found by adding the two numbers before it (1+2),
4.And the 5 is (2+3), and so on!
Example: the next number in the sequence above would be 21+34 = 55
It is that simple!

8. Mathematical Representation
In mathematical terms, the sequence Fn of Fibonacci numbers is
defined by the recurrence relation.
Fn = Fn-1 + Fn-2 Fn is term number "n"
Fn-1 is the previous term (n-1)
Fn-2 is the term before that (n-2)
seed values
F0 = 0, F1 = 1
in the first form, of
F1 = 1, F2 = 1
in the second form. Exam ple: term 9 would be calculated like
this:
F2 = 1, F3 = 2
F9 = F9-1 + F9-2
F9 = F8 + F7
34 = 13 + 21

9. Terms Below Zero
The sequence can be extended backwards! Like this:
They follow a +, -, +, -, ... pattern.
It can be written like this :
x−n = (−1)n+1 xn
Which says that term "-n" is equal to (−1)n+1 times term "n", and the value (−1)n+1
neatly makes the correct 1,-1,1,-1,... pattern.
n= ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ...
xn = ... -8 5 -3 2 -1 1 0 1 1 2 3 5 8 ...

10. HISTORY
The Fibonacci sequence appears in Indian
mathematics, in connection with Sanskrit
prosody.
In the early 1200’s, an Italian
mathematician Leonardo of Pisa
(nicknamed Fibonacci) discovered
the famous Fibonacci Sequence.
This sequence falls under the Mathematical domain of number theory and
its most famous problem concerned rabbits. The problem read:

12. Fibonacci
₴ Started his study, based on the
breeding habits of rabbits in 1202.
₴ He based his study in a set of ideal
Circumstances.

13. Fibonacci made the following assumptions
based on the breeding habits of rabbits
₴ He imagined a pair of rabbits in a field on
their own.
₴ One male AND one female.
₴ Rabbits never die.
₴ Female rabbits always produces one new
pair (one male, one female) every month
from the second month onwards.
₴ The gestation period for rabbits is one
month. (Not true.)

14. Activity 1
How many pairs will there be after 1
month ?
Mummy and Daddy have not yet
mated.

15. How many pairs of rabbits will there be after 2
months?
Mummy Daddy and their babies

16. How many pairs of rabbits will there be after 3
months?
Mummy, Daddy, their babies set 1 and their
babies set 2.

17. How many pairs of rabbits will there be after 4
months?
Mummy and Daddy, their three sets of babies and babies set 1 will now
produce their own set of babies.

18. How many pairs of rabbits will there be after 5
months?

19. How many pairs of rabbits will there be after 6
months?

20. 1, 1, 2, 3, 5, 8, 13, 21, 34

21. Activity 2
₴ Henry Dudeney (1857 – 1930)
₴ Adapted Fibonacci’s Rabbit problem to
cows.
₴ Only interested in Females.
₴ Changed months into years.
₴ Produced the same numbers, 1,1, 2, 3,
5, 8, 13, 21, 34, 55,,,,,,,

23. The Fibonacci
numbers are also an
example of a
complete sequence.
This means that every
positive integer can Specifically, every positive integer
can be written in a unique way as
be written as a sum of the sum of one or more distinct
Fibonacci numbers, Fibonacci numbers in such a way
where any one that the sum does not include any
number is used once two consecutive Fibonacci
at most. numbers. This is known as,
And a sum of Fibonacci
numbers that satisfies these
conditions is called a

24. 5
4
3

25. ₴ Starting with 5, every second Fibonacci number is the
length of the hypotenuse of a right triangle with integer
sides, or in other words, the largest number in a
Pythagorean triplet.
₴ The length of the longer leg of this triangle is equal to
the sum of the three sides of the preceding triangle.
₴ And the shorter leg is equal to the difference between the
preceding bypassed Fibonacci number and the shorter leg
of the preceding triangle.
₴ Example:
₴ The first triangle in this series has sides of length 5, 4,
and 3.
₴ Skipping 8, the next triangle has sides of length 13, 12
This(5 + 4 + 3), and 5 (8 − 3).
series continues
indefinitely. The triangle
sides a, b, c can be
calculated directly:

27. Two quantities are in Expressed algebraically:
the Golden Ratio if
a+b = a = φ
the ratio of the sum
of the quantities to a b
the larger quantity is
equal to the ratio of
the larger quantity to
the smaller one.
W here the Greek letter
phi ( Φ ) represents
the golden ratio. Its
value is:

28. An amazing finding concerning
the sequence is that the ratio of
two consecutive Fibonacci For example:
numbers approaches the golden
ratio or the golden number Phi. f(5)=5, f(4)=3
5/3 = 1.6666…
f(6)=8, f(5)=5
8/5=1.6
f(7)=13, f(6)=8
13/8=1.625
The values become closer and closer to the
golden number as the sequence continues,
which is a fascinating discovery.

30. • The Fibonacci spiral is
constructed by placing
together rectangles of
relative side lengths
equal to Fibonacci
numbers.
• A spiral can then be
drawn starting from the
corner of the first
rectangle of side length
1, all the way to the
corner of the rectangle of
side length 13.

31. Spiral Leaf
Growth

32. Plants can grow new cells
in spirals, such as the
pattern of seeds in this
beautiful sunflower.The
spiral happens naturally
because each new cell is
formed after a turn." N w e
c e ll, the n turn,
the n a no the r c e ll, the n
turn, . . . " How Far to Turn?
• So, if you were a plant, how much of a turn would you have in
between new cellS?
• if you don't turn at all, you would have a Straight line.
• but that iS a very poor deSign ... you want Something round
that will hold together with no gapS.

33. That is because the
Golden Ratio (1.61803...)
is the best solution to this
problem, and the
Sunflower has found this
solution in its own
natural way.
Because if you choose any
number that is a simple
fraction (example: 0.75 is
3/4, and 0.95 is 19/20, etc),
Why?
then you will eventually get
a pattern of lines stacking
up, and hence lots of gaps.

35. This interesting behavior is not just found in sunflower
seeds. Leaves, branches and petals can grow in spirals,
too.
Why?
So that new leaves don't block the sun from older leaves, or
so that the maximum amount of rain or dew gets directed
down to the roots.
In fact, if a plant has spirals, the rotation tends to be a fraction
made with two successive Fibonacci Numbers.

36. For Example:
• A half rotation is 1/2 (1 and 2 are Fibonacci Numbers)
• 3/5 is also common (both Fibonacci Numbers), and
• 5/8 also
all getting closer and closer to the Golden Ratio.
And that is why Fibonacci Numbers are very common in plants.
1,2,3,5,8,13,21,... etc occur in an amazing number of places.
CABBAGE
PINECONE

37. • Music involves several
applications of Fibonacci
numbers.
• A full octave is composed
of 13 total musical tones,
8 of which make up the
actual musical octave.

38. • An interesting use of
the Fibonacci sequence !
is for converting miles to iles
re m w
kilometers. 5 mo ho
• For instance, if you But h in
want to know how many uc rs?
m ete
kilometers 8 miles is.
• Take the Fibonacci kilo m
number (8) and look at
the next one (13). 8
miles is about 13
kilometers.
• This works because it so
happens that the
conversion factor
between miles and
kilometers (1.609) is
roughly equal to
(1.618).

39. The Fibonacci
Sequence has also
been linked to the
human face and Each part of the
hands. index finger,
beginning from the
tip down to the wrist,
is larger than the
preceding section by
about the ratio of
1.618 (the golden
ratio) (Human
The FibonacciHand).
sequence can also be found in the
structure of the human hand.

40. The Golden (or Divine)
Ratio has been talked
about for thousands of
years.
1.618
People have shown that all
things of great beauty
have a ratio in their
dimensions of a number 1
Leonardo da Vinci’ s
around 1.618.
painting of the Mona Lisa
is based on the
arrangement of the golden
rectangle. Many argue that
this is the reason behind
its beauty. Mona Lisa’s painting has Golden Ratio
applicable to it.

41. M any famous artists, for
example, have used
golden rectangles in the
structure of their
artwork. L eonardo da
Vinci showed that in a
‘perfect man’ there were
lots of measurements
that followed the Golden
Ratio.

42. There is much more to be
discovered.
Hence………
The story continues.