Maths Investigation: Pascals Triangles

Maths investigation

John Abarshi

Part 1: Rows Pattern

From the first 5 rows, I observe a pattern which is repeated on the subsequent row. First, the number 1 stands at the pivot of the pyramid and begins and ends each row. Second, the numbers follow a sequence 1,2,3,….both diagonally and downwards, and are same per row on the right and left sides of the pyramid, and the first number coincides with the number of the row. A similar sequence follows next (1,3,6..) on both sides, and then (1,4,10…) etc. Third, every number on each row (excluding the 1’s) is the sum of the two numbers sited immediately above it. Four, on every other row (i.e. the even number rows), there is a centrally-located number (2,6,20..), and this also has a sequence; it corresponds to the sum of the same number (1+1,3+3,10+10). The rows that fall in between consist of and even number of digits (i.e. no middle number here), but the two middle numbers are always the same (1,1; 3,3; 10,10;..). Therefore you have a small pyramid which is exactly symmetrically shaped in the middle such that a sequence shows every time a new row is formed. You can see the pattern 2,3,4. I can predict that the next one will be five and the one after six and so on. Each number in the triangle is the sum of the two directly above it. So the next row will be. 1, 5, 10, 10, 5, 1. this is because you put 1 first then add 1 + 4 which is 5, then you add 4 + 6 which is 10 and 4 + 6 again. then you 1+4 again and you add a 1 to the outside. The next row will be. 1, 6, 15, 20, 15, 6, 1. This is the seventh row. And the eight row will be 1, 7, 21, 35, 35, 21, 7, 1. You can clearly see a pattern occurring. You always add up the two numbers above together and this is how you will be able to find out what the next row is. See below:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1