Pascal's Triangle2

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Pascal's Triangle
Pascal2.gif
Field: Algebra
Image Created By: Math Forum
Website: Mathforum.org

Pascal's Triangle

Pascal's Triangle


Basic Description

Pascal's triangle is a triangular arrangement of specific numbers which have interesting patterns. We start out with 1 and 1, 1, for the first two rows. To construct each entry in the next row, we sum the two numbers immediately above it to the right and to the left. If there's only one number diagonally above the entry to the left and to the right, then we enter just that number. We can continue doing this endlessly. This pattern is just one of the many patterns within the triangle.

A More Mathematical Explanation

Each entry in a Pascal's triangle is identified by a row and a place. The rows are labeled starting f [...]

Each entry in a Pascal's triangle is identified by a row and a place. The rows are labeled starting from zero, so the first row would be row 0, the second one would be row 1, the third, row 2 and so forth. Places are given to each entry starting from the first number after 1, from the left to the right. 70 for example would be identified as row 8, place 4.

Properties

  • The triangle is bordered by 1's on the right and left edges.
  • The next line of numbers in diagonal order after the edge numbers are natural numbers 1,2,3,4...
  • The next set of numbers inwards after the natural numbers are triangular numbers 1,3,6,10,15...
  • After the triangular numbers we have tetrahedral numbers in order 1,4,10,20...
  • The next d-diagonal contains the next higher dimensional "d-simplex" numbers.
  • The first number after 1 in each row divides all the other numbers in that row if and only if it is prime.

Patterns within Pascal's triangle

Pascal's triangle contains a number of smaller patterns within it. Some of these patterns include:


Magic 11's

If a row is made into a single number (omitting the spaces), the resulting number is equal to 11 raised to the n^th power, where n is the row number. For example:

Row # Actual Row Single Number Formula
0 1 1 11^0
1 1 1 11 11^1
2 1 2 1 121 11^2
3 1 3 3 1 1,331 11^3
4 1 4 6 4 1 14,641 11^4
5 1 5 10 10 5 1 15,101,051 11^5
6 1 6 15 20 15 9 1 1,615,201,561 11^6

Hockey Stick Pattern

Hockey stick pattern http://ptri1.tripod.com/

If we select entries diagonally starting from 1 on the edge up to any point in the triangle, the sum of the numbers selected is equal to the number below the selection that is not on the diagonal. To understand this better, please refer to the tutorial on the below.

Fibonacci Sequence

The Fibonacci sequence is a list of Fibonacci numbers. Each Fibonacci number is generated by adding the previous two consecutive terms in the sequence. The first two Fibonacci numbers are 0 and 1. This pattern can be located in Pascal's triangle. The sum of the entries in each row, as shown in the tutorial gives a Fibonacci number. This creates the sequence 1,1,2,3,5,8,13,21,34,...which is the Fibonacci sequence.









Sierpenski Triangle

Sierpenski triangle generated by coloring odd and even numbers with Pascal's triangle with two different colors http://en.wikipedia.org/wiki/Pascal's_triangle

If we are to color the odd and even numbers in Pascal's triangle with two distinct colors, we would observe an interesting recursive pattern seen in the Sierpinski triangle. You could try this out in the tutorial and see what happens.

These are just a few of the patterns observed in Pascal's triangle. Other patterns include:

Animation Demonstrating Patterns

The Flash animation below allows you to explore some of the patterns present in Pascal's Triangle:

Applications of Pascal's Triangle

In Algebra

Pascal's triangle can be used to determine binomial coefficients in binomial expansions. For example

(x + y)^2 = x^2 + 2xy + y^2 =1{x^2}{y^0} + {2x^1}{y^1} +{1x^0}{y^2}

In the above example, notice that the coefficients in the expansion are exactly the same as the entries in row two in Pascal's triangle.

This is summarized in the binomial theorem which states that in general,

(x + y)^n = {a_0}{x^n} + {a_1}{x^{n-1}} + {a_2}{x^{n-2}}{y^2}++ {a_{n-1}}{xy^{n-1}} + {a_n}{y^n}


where a_i are the entries in row n in Pascal's triangle.

In Combinatorics




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