StUdEnT = Anushka Sahu

StAnDaRd = Tenth ‘ A ’

EnRoLlMeNt = 10015

SuBjEcT = Mathematics

ToPiC = FIBONACCI SEQUENCE

QUESTIONAIRRE1. 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

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

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)

6. Complete the lyrics for the famous Christmas song, "The Twelve Days of Christmas": "___ maids a-milking..."

• 8• 10• 6• 9

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

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)

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• -18. The popular card game 'Blackjack', known as 'Pontoon' in the UK, is also called ___.

• 17• 70• 3• 219. 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

ANSWERS1)1

2)1

3)2

4)3

5)5

6)8

7)13

8)21

9)34

10) 55

These Answers form a Fibonacci Series !

These Answers form a Fibonacci Series !

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!

Mathematical Representation

Fn = Fn-1 + Fn-2

seed valuesF0 = 0, F1 = 1

in the first form, ofF1 = 1, F2 = 1

in the second form.F2 = 1, F3 = 2

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation.

Fn is term number "n" Fn-1 is the previous term (n-1) Fn-2 is the term before that (n-2)

Fn is term number "n" Fn-1 is the previous term (n-1) Fn-2 is the term before that (n-2)

Example: term 9 would be calculated like this:

F9 = F9-1 + F9-2F9 = F8 + F734 = 13 + 21

Example: term 9 would be calculated like this:

F9 = F9-1 + F9-2F9 = F8 + F734 = 13 + 21

Terms Below ZeroThe sequence can be extended backwards! Like this:

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 ...

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.

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:

Fibonacci

₴ Started his study, based on the breeding habits of rabbits in 1202.

₴ He based his study in a set of ideal Circumstances.

₴ 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.)

Fibonacci made the following assumptionsbased on the breeding habits of rabbits

Activity 1

How many pairs will there be after 1 month ?

Mummy and Daddy have not yet mated.

How many pairs of rabbits will there be after 2 months?

Mummy Daddy and their babies

How many pairs of rabbits will there be after 3 months?

Mummy, Daddy, their babies set 1 and their babies set 2.

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.

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

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

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

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,,,,,,,

Specifically, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as,

The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.

And a sum of Fibonacci numbers that satisfies these conditions is called a

54

3

₴ 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 (5 + 4 + 3), and 5 (8 − 3). This series continues indefinitely. The triangle sides a, b, c can be calculated directly:

Two quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.

Where the Greek letter phi ( Φ ) represents the golden ratio. Its value is:

Expressed algebraically:

a+b = a = φ a b

An amazing finding concerning the sequence is that the ratio of two consecutive Fibonacci numbers approaches the golden ratio or the golden number Phi.

The values become closer and closer to the golden number as the sequence continues, which is a fascinating discovery.

For example:

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 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.

Spiral Leaf Growth

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."New cell, then turn, then another cell, then 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.

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), then you will eventually get a pattern of lines stacking up, and hence lots of gaps.

Why?

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.

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

• 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.

• An interesting use of the Fibonacci sequence is for converting miles to kilometers.

• For instance, if you want to know how many kilometers 8 miles is.

• Take the Fibonacci 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).

5 more miles!

But how

much in

kilometers?

The Fibonacci Sequence has also been linked to the human face and hands.

The Fibonacci sequence can also be found in the structure of the human hand.

Each part of the 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 Hand).

The Golden (or Divine) Ratio has been talked about for thousands of years.

People have shown that all things of great beauty have a ratio in their dimensions of a number around 1.618.

1

1.618

Leonardo da Vinci’s 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.

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

There is much more to be discovered.

Hence………

The story continues.