MAT 214 Prof. Erstenyuk Name: 1.2 Patterns and Pascal's Triangle Algebra note: A linear equation (which graphs a line) with x as the highest powered term, will have one row of successive differences which will remain the same each time - that difference is the slope of the line. A quadratic equation (which graphs a parabola) with x2 as the highest powered term, will have two rows of successive differences before we get a difference that stays the same each time. An equation with x' as the highest powered term, will have three rows of successive differences before the difference stays the same each time. Part II: Number Patterns in Pascal's Triangle a.) Diagonals The first diagonal in the triangle (marked by the arrow) is all 1's. The diagonal next to it is the integers (1, 2, 3, etc.). What special figurate numbers are found in the third diagonal (the one that starts (1, 3, 6...)? 15 20 15 35 35 56 70 56 28 84 126 126 84 136 120 210 252 210 120 45 10 165 330 462 462 330 [165 155 220 495 792 924 792 495 | 220 66 12 715 1287 1716 1716 1287 715 1 286 78 13 1001 2002 3003 3432 3003 2002 1001 ) 364 [91 14 105 1365 3003 5005 6435] 6435 5005 3003 1365 455 10S b.) The powers of 11 are 110 = 1, 11' = 11, 112 = , 113 = 114 = Where do you see the powers of 11 in Pascal's triangle? Hint: don't look for each answer to come up in one number, look at rows of numbers. Does the pattern keep going on after 114? Explain