pascal's triangle by adarsh tiwari ,kv andrewsgang, class 7 a
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PASCAL'S TRIANGLEAND ITS APPLICATIONS
Adarsh Tiwari Class- VII-A
Kendrya Vidyalaya Andrews Ganj ,New Delhi-24
1Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Pascal’s Triangle
Introduction Pascal Triangle Patterns Applications
2Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
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
3Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Pascal's 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 Pascal's Triangle
4Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Pascal's Triangle
5Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Pascal's Triangle
6Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Fibonacci SeriesFrom Pascal Triangle
7Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
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 redline , it results in series called Fibonacciseries . 1,1,2,3,5…….
8Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
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 andbuilding in ancient times.
It is known as phi / Φ
9Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
SpiralsFrom
Pascal Triangle
We see spirals aroundus in shells, galaxies,etc.
This is also drawnwith Fibonacci series.1,1,2,3,5,8,13……….
10Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
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 anysymbol or colors, to get “SierpinskiTriangle”.
11Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Use of Power in Pascal's 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 Pascal's triangle
12Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Use of Power in Pascal's 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 Pascal's triangle
13Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Summing The RowsSumming The Rows
11
1 1 ++ 11
1 1 ++ 2 2 ++ 11
1 1 ++ 3 3 ++ 3 3 ++ 11
1 1 ++ 4 4 ++ 6 + 4 + 16 + 4 + 1
1 1 ++ 5 5 ++ 10 10 ++ 10 10 ++ 5 5 ++ 11
1 1 ++ 6 6 ++ 15 15 ++ 20 20 ++ 15 15 ++ 6 6 ++ 11
=1=1
=2=2
=4=4
=8=8
=16=16
=32=32
=64=6414Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
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 Pascal's Triangle.
15Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Pascal’s Triangle: Row Binomial coefficients of (1+X)0 (1+X)1 , (1+X)2
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
(1+X)0 = 1
(1+X)1 = 1+1X
(1+X)2 =
(1+X)3 =1 + 3X + 3X2 + 1X3
(1+X)4 =1 + 4X + 6X2 + 4X3 + 1X4
1 + 2X + 1X2
16Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Hockey Stick Pattern
17Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Hockey Stick Pattern
The dark numberslooks like hockey stick.
To draw Hockey stickadd the numbers of thelonger line , summationis the left number.
example- 1+2=3 or1+1+1+1=4
18Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Symmetry Pascal's Triangle You must be familiar with this word
``symmetry”. See symmetry in Pascal's triangle.
19Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Symmetry Pascal's Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
20Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
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.
21Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
Any Questions?
22Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12
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