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Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
Pascal's triangle [compatibility mode]
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Pascal's triangle [compatibility mode]

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  • 1. PASCALS TRIANGLEAND ITS APPLICATIONS Adarsh Tiwari Class- VII-AKendrya Vidyalaya Andrews Ganj ,New Delhi-24 Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 1
  • 2. Pascal’s Triangle Introduction Pascal Triangle Patterns Applications Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 2
  • 3. Blaise Pascal French Mathematician born in 1623 At the age of 19, he invented one of the first calculating machines which actually worked. It was called the Pascaline Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 3
  • 4. Pascals Triangle What is a Pascal’s triangle? Pascal triangle is algebraic pattern. It was invented by Blaise Pascal. There are many algebraic patterns like hockey stick pattern, spiral, and Sierpinski triangle etc. in Pascals Triangle Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 4
  • 5. Pascals TriangleAdarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 5
  • 6. Pascals TriangleAdarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 6
  • 7. Fibonacci Series From Pascal TriangleAdarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 7
  • 8. Fibonacci Series In this series the next term is addition of previous two numbers. the Red line passing through Pascal Triangle, by addition of the terms of red line , it results in series called Fibonacci series . 1,1,2,3,5……. Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 8
  • 9. Golden Ratio /Number Fibonacci Series is 1,1,2,3,5,8,13 Golden number is Ratio between two adjacent terms of Fibonacci series. Golden ratio(example 8/5=1.6) Example of this ratio we get in natural Growth like bone growth, plant growth and building in ancient times. It is known as phi / Φ Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 9
  • 10. Spirals From Pascal Triangle We see spirals around us in shells, galaxies, etc. This is also drawn with Fibonacci series. 1,1,2,3,5,8,13………. Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 10
  • 11. Sierpinski Triangle From Pascal Triangle  From Pascal Triangle we can draw Sierpinski triangle.  I have used O for the even numbers and I for the odd numbers . You can use any symbol or colors, to get “Sierpinski Triangle”. Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 11
  • 12. Use of Power in Pascals Triangle Power of 2  First: (2)0 =1  Second: (2)1 =2  Third: (2)2 =4  Forth: (2)3 =8  Look at the result, they are the sum of each row of the Pascals triangle Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 12
  • 13. Use of Power in Pascals Triangle Power of 11  First: (11)0 =1  Second: (11)1 =11  Third: (11)2 =121  Forth: (11)3 =1331  Look at the result, they are the terms combined together of the Pascals triangle Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 13
  • 14. Summing The Rows1 =11 + 1 =21 + 2 + 1 =41 + 3 + 3 + 1 =81 + 4 + 6 + 4 + 1 =161 + 5 + 10 + 10 + 5 + 1 =321 + 6 + 15 + 20 + 15 + 6 + 1 =64  Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12  14
  • 15. Binomial Coefficient (a+b)*(a+b)=1a*a+2a*b+1b*b The numbers which are colored with red are same as the number in the 3rd row of the Pascals Triangle. Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 15
  • 16. Pascal’s Triangle: Row Binomial coefficients of (1+X)0 (1+X)1 , (1+X)2 (1+X)0 = 1  1 (1+X)1 = 1+1X  1 1 (1+X)2 = 1 + 2X + 1X2  1 2 1 (1+X)3 =1 + 3X + 3X2 + 1X3  1 3 3 1 (1+X)4 =1 + 4X + 6X2 + 4X3 + 1X4  1 4 6 4 1 16 Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
  • 17. Hockey Stick PatternAdarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 17
  • 18. Hockey Stick Pattern  The dark numbers looks like hockey stick.  To draw Hockey stick add the numbers of the longer line , summation is the left number.  example- 1+2=3 or 1+1+1+1=4Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 18
  • 19. Symmetry Pascals Triangle You must be familiar with this word ``symmetry”. See symmetry in Pascals triangle. Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 19
  • 20. Symmetry Pascals Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 11 6 15 20 15 6 1 Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 20
  • 21. Application Pascal triangle is algebraic pattern. From it we make many pattern like Serpenski Triangle , hockey stick pattern ,etc. Fibonacci series , 1, 1, 2, 3, 5, 8, can be seen in the growth in animals plants , shells & spirals. Olden Greece buildings used Golden Ratio . Binomial coefficients from Pascal Triangle . Square numbers 1, 4, 6, 25, 36...... Counting numbers 1, 2, 3, 4, 5, ...... Triangular numbers 1, 3, 6, 10, 15........ Powers of two 1, 2, 4, 8, 6........ Probability and Games from Pascal Triangle. Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 21
  • 22. Any Questions? Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 22

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