Pascal’s Triangle MDM 4U Lesson 5.1

PASCAL’S TRIANGLE: -originated in China about 1100 AD and it was developed by Blaise Pascal. Chinese Pascal’s Triangle Japanese Pascal’s Triangle

Pascal’s Triangle The first 9 rows of Pascal’s triangle:

Properties of Pascal’s Triangle (just a few) Any number in the interior of Pascal’s triangle will be the sum of the two numbers appearing above it. The first row is referred to as “row 0”. The sum of the rows are 1, 2, 4, 8, 16, 32, etc (2n) The numbers in the rows follow the pattern of base 11.

We will use the theories of COMBINATIONS from chapter 4 to make a version of Pascal’s triangle we can use and understand. Arrange nCr in a triangle. Value of n 1 2 3 4 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

In any row of Pascal’s Triangle, entries equidistant from each end are equal. For example, C(5,0) = 1 and C(5,5) = 1. This can be expressed generally as:

Pascal’s Identity Another very important discovery made by Pascal, is the relationship between the sum of consecutive values in one row and the value found in the next row immediately beneath. The pattern discovered in general form is:

Example 1 a) Find the sums of the numbers in each of the first six rows of Pascal’s triangle. b) Now predict the sum of the numbers in rows 7, 8, 9. c) Predict the sum of the entries in row n of Pascal’s triangle.

Example 2 a) The first 6 terms of row 25 of Pascal’s triangle are 1, 25, 300, 2300, 12650, 53130. Determine the first 6 terms in row 26.

Example 3 a) Which row in Pascal’s Triangle has the sum of its terms equal to 32768?

Example 4 Can you find a relationship between perfect squares and the sums of pairs of entries in Pascal’s triangle?

Practice Questions Page 251 #1, 3, 4, 7, 8, 10, 16 (you really want to get these done before class tomorrow or you will be confused when we start to apply Pascal’s triangle….)