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# Fibonacci sequence

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### Fibonacci sequence

1. 1. StUdEnT = Anushka Sahu StAnDaRd = Tenth ‘ A ’ EnRoLlMeNt = 10015SuBjEcT = Mathematics ToPiC = FIBONACCI SEQUENCE
2. 2. QUESTIONAIRRE1. Lets 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• 172. 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. Evennumbers are specifically those numbers which, when divided by ___, give aremainder of zero.• 1• 2• 7• 42009
3. 3. 4. A duo is a group of *two* people or things. For example, Batman and Robinare 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• 56. Complete the lyrics for the famous Christmas song, "The Twelve Days of Christmas": "___ maids a-milking..."• 8• 10• 6• 9
4. 4. 7. This number is considered unlucky by most cultures, but is lucky for traditional Chinesepeople. At the Last Supper, this was the number of people (including Christ) who were present.A bakers 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 atomicnumber?• Sir Humphrey Appleby• 65,536• 34• 110. Now, if youve 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)
5. 5. ANSWERS1) 12)13)2 These Answers4)35)5 form a Fibonacci6)8 Series !7)138)219)3410) 55
6. 6. 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 = 55It is that simple!
7. 7. Mathematical RepresentationIn mathematical terms, the sequence Fn of Fibonacci numbers isdefined 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 valuesF0 = 0, F1 = 1in the first form, ofF1 = 1, F2 = 1in 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
8. 8. Terms Below ZeroThe sequence can be extended backwards! Like this:They follow a +, -, +, -, ... pattern.It can be written like this :x−n = (−1)n+1 xnWhich says that term "-n" is equal to (−1)n+1 times term "n", and the value (−1)n+1neatly 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 ...
9. 9. HISTORY The Fibonacci sequence appears in Indianmathematics, in connection with Sanskritprosody. 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 andits most famous problem concerned rabbits. The problem read:
10. 10. Fibonacci₴ Started his study, based on the breeding habits of rabbits in 1202.₴ He based his study in a set of ideal Circumstances.
11. 11. 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.)
12. 12. Activity 1How many pairs will there be after 1 month ? Mummy and Daddy have not yet mated.
13. 13. How many pairs of rabbits will there be after 2 months? Mummy Daddy and their babies
14. 14. How many pairs of rabbits will there be after 3 months? Mummy, Daddy, their babies set 1 and their babies set 2.
15. 15. 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.
16. 16. How many pairs of rabbits will there be after 5months?
17. 17. How many pairs of rabbits will there be after 6months?
18. 18. 1, 1, 2, 3, 5, 8, 13, 21, 34
19. 19. 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,,,,,,,
20. 20. The Fibonaccinumbers are also anexample of acomplete sequence.This means that everypositive integer can Specifically, every positive integer can be written in a unique way asbe written as a sum of the sum of one or more distinctFibonacci numbers, Fibonacci numbers in such a waywhere any one that the sum does not include anynumber is used once two consecutive Fibonacciat most. numbers. This is known as, And a sum of Fibonacci numbers that satisfies these conditions is called a
21. 21. 54 3
22. 22. ₴ 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, 12This(5 + 4 + 3), and 5 (8 − 3). series continuesindefinitely. The trianglesides a, b, c can becalculated directly:
23. 23. 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:
24. 24. 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.
25. 25. • 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.
26. 26. Spiral LeafGrowth
27. 27. 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 dont 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.
28. 28. 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 anynumber that is a simplefraction (example: 0.75 is3/4, and 0.95 is 19/20, etc), Why?then you will eventually geta pattern of lines stackingup, and hence lots of gaps.
29. 29. This interesting behavior is not just found in sunflower seeds. Leaves, branches and petals can grow in spirals, too. Why?So that new leaves dont 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.
30. 30. 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
31. 31. • 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.
32. 32. • 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).
33. 33. 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.
34. 34. The Golden (or Divine)Ratio has been talkedabout for thousands ofyears. 1.618People have shown that allthings of great beautyhave a ratio in theirdimensions of a number 1Leonardo da Vinci’ saround 1.618.painting of the Mona Lisais based on thearrangement of the goldenrectangle. Many argue thatthis is the reason behindits beauty. Mona Lisa’s painting has Golden Ratio applicable to it.
35. 35. M any famous artists, forexample, have usedgolden rectangles in thestructure of theirartwork. L eonardo daVinci showed that in a‘perfect man’ there werelots of measurementsthat followed the GoldenRatio.
36. 36. There is much more to be discovered.Hence………The story continues.