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Pascal's triangle [compatibility mode]
1. PASCAL'S TRIANGLE
AND ITS APPLICATIONS
Adarsh Tiwari
Class- VII-A
Kendrya 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. 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
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 4
7. Fibonacci Series
From Pascal Triangle
Adarsh 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 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
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 12
13. 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
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 13
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 Pascal's Triangle.
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 15
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=4
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 18
19. Symmetry Pascal's Triangle
You must be familiar with this word
``symmetry”.
See symmetry in Pascal's triangle.
Adarsh Tiwari , Class 7 -A, KV Andrews Ganj N Delhi-24 , Aug 12 19
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