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PASCAL'S TRIANGLEAND ITS APPLICATIONS
Adarsh Tiwari Class- VII-A
Kendrya Vidyalaya Andrews Ganj ,New Delhi-24
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Pascal’s Triangle
Introduction Pascal Triangle Patterns Applications
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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
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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
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Pascal's Triangle
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Pascal's Triangle
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Fibonacci SeriesFrom Pascal Triangle
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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…….
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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
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SpiralsFrom
Pascal Triangle
We see spirals aroundus in shells, galaxies,etc.
This is also drawnwith Fibonacci series.1,1,2,3,5,8,13……….
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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”.
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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
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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
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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
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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.
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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
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Hockey Stick Pattern
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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
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Symmetry Pascal's Triangle You must be familiar with this word
``symmetry”. See symmetry in Pascal's triangle.
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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
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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.
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Any Questions?
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