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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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