Pascal Triangle
Download as DOCX, PDF
0 likes58 views
1. The Pascal triangle is a triangular array where each number is the sum of the two numbers directly above it. Patterns emerge such as the first diagonal containing all 1s, the second diagonal containing counting numbers, and the third diagonal containing triangular numbers. 2. The sums of each horizontal row are powers of 2, following the pattern 20 = 1, 21 = 2, 22 = 4, etc. Each line is also a power of 11, such that 110 = 1, 111 = 11, 112 = 121, and so on. 3. For the second diagonal, the square of each number equals the sum of the two numbers above it and below. For example, 32 = 3 + 6 =
Pascal Triangle
- 1. PASCAL TRIANGLE (Reymark J. Lubon) One of the most interesting Number Patterns is Pascals Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher). Starting Point of the Investigation: 1. Build a triangle, start with 1 at the top, and then continue placing numbers below it in a triangle pattern. Each number is the number directly above it added together. What have you observe? As Ive observe the first diagonal are compose of 1 only, the second diagonal is a counting numbers, the third diagonal is triangular numbers and the fourth diagonal has the tetrahedral numbers. Another observation was that they are Symmetric. 2. Continue placing numbers below it and notice about the horizontal sums? Is there a pattern? Make a conjecture out of it. I observe that they double each time. So therefore, my first conjecture was Conjecture No. 1: Each horizontal sum is a power of 2 denoted by . Proof: 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, 25 = 32 . 1 1 2 1 1 1 33 11
- 2. 3. Notice that each line is also the powers of 11. 110 = 1 ( ≠ "1") 111 = 11 ( ≠ "1" and "1") 112 = 121 ( ≠ ≠ "1" , "2" and "1") But what happens with 115 , 116 , ＄.? Make a conjecture about it. What happens with 115 ? Simple! The digits just overlap, and then add the two digits like this: 115 = So, the final answer is 161051. 116 = So, the final answer is 1771561 Conjecture No. 2: The horizontal numbers is a power of 11 denoted by . 4. For the second diagonal, what have you notice? What is the relation between the sums of the two numbers below it? Construct a conjecture. For the second diagonal, the square of a number is equal to the sum of the numbers next to it and below both of those. Examples: 32 = 3 + 6 = 9 42 = 6 + 10 = 16 52 = 10 + 15 = 25 Conjecture No. 3: The square of a number is equal to the sum of the numbers next to it and below both of those. 1 51 01 0 5 1 1 6 1 0 5 1 61 52 01 5 61 1 1 7 7 1 5 6 1