Pascal's triangle and fibonacci
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This document discusses Pascal's triangle and its relationship to the Fibonacci sequence. It explains that Pascal's triangle is an ancient mathematical pattern where each number is the sum of the two numbers above it. The diagonals of the triangle relate to important numerical sequences like the counting numbers, triangular numbers, and the Fibonacci sequence. The document also outlines various properties of Pascal's triangle, such as the horizontal rows summing to powers of two and representing powers of eleven. It provides examples of how Pascal's triangle can be used to solve probability and binomial expansion problems.
Pascal's triangle and fibonacci
- 1. PASCAL S TRIANGLE AND FIBONACCI ΔΙΑΘΕΜΑΤΙΚΗ ΔΡΑΣΗ: ΑΓΓΛΙΚΑ +ΜΑΘΗΜΑΤΙΚΑ ΤΜΗΜΑ Γ3 Π.Π. ΓΥΜΝΑΣΙΟ ΙΩΝΙΔΕΙΟΥ ΣΧΟΛΗΣ ΠΕΙΡΑΙΑ 2014 2015 Ιωάννα Λύρα Βίκτωρ Μαρτίνος Μυρτώ Φιλιππάκη Βασίλης Μιχαλίτσης Σακκάς Γιώργος Μαρίνης Στέλιος
- 2. HISTORY Pascal s triangle is perhaps the most famous of all number patterns. It is very ancient standing, being probably more than a thousand year old. Its hidden properties have been revealed more and more as mathematics has developed through the centuries. The Chinese Knew About It This drawing is entitled "The Old Method Chart of the Seven Multiplying Squares". It is from the front of Chu Shi- Chieh's book "Ssu Yuan Yü Chien" (Precious Mirror of the Four Elements), written in AD 1303 (over 700 years ago, and more than 300 years before Pascal!), and in the book it says the triangle was known
- 3. CONSTRUCTION To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. Each number is the two numbers above it added together (except for the edges, which are all "1").
- 4. THEORY Pascal’s Triangle, developed by the French Mathematician Blaise Pascal, is formed by starting with an apex of 1. Every number below in the triangle is the sum of the two numbers diagonally above it to the left and the right, with positions outside the triangle counting as zero. The numbers on diagonals of the triangle add to the Fibonacci series, as shown below:
- 5. PATTERNS WITHIN THE TRIANGLE Diagonals The first diagonal is, of course, just "1"s, and the next diagonal has the Counting numbers(1,2,3, etc). The third diagonal has the triangular numbers (The fourth diagonal, not highlighted, has the tetrahedral numbers.)
- 6. FIBONACCI SEQUENCE The Fibonacci Sequence starts "0, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc
- 7. PROPERTIES Horizontal rows add to powers of 2 (i.e., 1, 2, 4, 8, 16, etc.) The horizontal rows represent powers of 11 (1, 11, 121, 1331, 14641) for the first 5 rows, in which the numbers have only a single digit. Adding any two successive numbers in the diagonal 1- 3-6-10-15-21-28… results in a perfect square (1, 4, 9, 16, etc.) It can be used to find combinations in probability problems (if, for instance, you pick any two of five items, the number of possible combinations is 10, found by looking in the second place of the fifth row. Do not count the 1’s.) When the first number to the right of the 1 in any row is a prime number, all numbers in that row are divisible by that prime number
- 8. Pascal's Triangle can also show you the coefficients in binomial expansion
- 9. SOURCES http://www.mathsisfun.com/pascals-triangle.html http://www.goldennumber.net/pascals-triangle/ H.E. HUNTLEY, The divine proportion, Dover Publications, N.York 1970 Α.ΑΛΙΜΠΙΝΙΣΗΣ, Σ.ΓΡΗΓΟΡΙΑΔΗΣ, Ε.ΕΥΣΤΑΘΟΠΟΥΛΟΣ, Ν.ΚΛΑΟΥΔΑΤΟΣ, Σ.ΠΑΠΑΣΤΑΥΡΙΔΗΣ, Α.ΣΒΕΡΚΟΣ , ΜΑΘΗΜΑΤΙΚΑ Γ΄ ΓΥΜΝΑΣΙΟΥ . ΟΕΔΒ,ΑΘΗΝΑ 2004