Pascal's triangle by Adarsh Tiwari ,KV Andrewsgang, Class 7 A

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

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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