Maths Investigation: Pascals Triangles

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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.