Originally Pascal’s Triangle was developed by the Chinese of long ago. But then the French mathematician Blaise Pascal was officially the first person to discover the importance of the patterns it had within itself. But how exactly does it work??? In this research paper, I will explain how to make the Pascal’s Triangle and why it is so special. Construction: Pascal’s Triangle is basically a triangle of numbers. “At the tip of the triangle is the number 1, which makes up row zero. Then the second row has two 1’s by adding the 2 numbers above them to the left and right, 1 and 0 (all numbers outside the triangle are zeros).
Now do the same for the second row. ” 0+1= 1, 1+2=3, 2+1=3, 1+0=1. Then the results become the third row. 0+1=1, 1+3=4, 3+3=6, 3+1=4, and 1+0=1. Then the pattern continues on infinitely. ? There seems to be many patterns in this triangle. For example: The Sums of the Rows. The Sums of the Rows: “The sum of the numbers in any row is equal to 2 to the nth power or 2^n when n represents the number of the row. ” For example: 20 = 1 21 = 1+1 = 2 22 = 1+2+1 = 4 23 = 1+3+3+1 = 8 24 = 1+4+6+4+1 = 16 Prime Numbers: If the 1st element in a row is a prime number (remember, the 0th element of every row is 1), all the numbers in that row (excluding the 1’s) are divisible by it. For example, in row 7 (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7. ” The Hockey Stick: “If a diagonal of numbers of any length is selected starting at any of the 1’s bordering the sides of the triangle and ending on any number inside the triangle on that diagonal, the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself. Look at the example on the next page if you don’t understand. ? Magic 11’s: “If a row is made into a single number by using each element as a digit of the number (carrying over when an element itself has more than one digit), the number is equal to 11 to the nth power or 11n when n is the number of the row the multi-digit number was taken from. ” Row #Formula=Multi-Digit numberActual Row Row 0110=11 Row 1111=111 1 Row 2112=1211 2 1 Row 3113=13311 3 3 1 Row 4114=146411 4 6 4 1 Row 5115=1610511 5 10 10 5 1
Row 6116=17715611 6 15 20 15 6 1 Row 7117=194871711 7 21 35 35 21 7 1 Row 8118=2143588811 8 28 56 70 56 28 8 1 Connection to Sierpinski’s Triangle: When all the odd #s in the triangle are filled in and the rest are left blank, the Sierpinki Triangle is revealed again showing another pattern in the Pascal’s triangle. ? Pascal’s Triangle is somewhat difficult because of some of the patters that appear. Patterns like Fibonnacci’s Sequence and Polygonal Numbers are hard to make out because there are many formulas involved.
That is the reason why I am not going into those patterns much. But the triangle goes on forever and ever and that makes the patterns continuous as well. Overall Pascal’s Triangle is very complex and should not be taken lightly because of its simple structure. But sadly this is the end of my report and I hoped you learned something. BIBLIOGRAPHY http://www. mathsisfun. com/pascals-triangle. html http://www. mu6. com/catalan_numbers_growth. html http://ptri1. tripod. com/