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Math exploration of Pascals Triangle

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Blaise Pascal wrote his work titled “Treatise on the Arithmetical Triangle” in 1654 but was published until 1665. This work had an immense role in the development of the probability theory, theory of convergent and divergent series, derivative and integral calculus. Derivation of Pascal’s Triangle Consider the following binomial expansions in Figure 1 below. It is conspicuously evident that the coefficients of binomial expansions of successive degrees by using the Pascal’s triangle. The numbers in each row except the one at the apex of the triangle can be attained by adding two underlying adjacent numbers within the aforementioned row.

It true even for the numbers bordering the triangle if there is assumption that there are invisible zeros that extend to the sides of each row There are several observations that can be from the Pascal’s triangle.

The first observation is that all the numbers are positive. Only positive numbers can be generated including additional 1’s or adding existing positive numbers. There exists a vertical line within the symmetry through the apex of the triangle. The identical thing is undertaken on both sides of the line of symmetry, and therefore the same results are obtained (Cullinane, pp. 145-178). Moreover, it is in agreement with the fundamental idea in mathematics where if you do the same thing to the same subjects, the same results. Rows that are parallel to the edges of the triangle also depict interesting patterns.

For example 1, 3, 6, 10, 15 and so on are just the sums of (1), (1+2), (1+2+3), (1+2+3+4) and so on. Moreover, it is swift elaboration of generations’ methods of the numbers, and sometimes referred to as triangular numbers because they are generated by an equilateral triangle (Bassarear, pp178-212). Alternation of the signs of the numbers in any row and adding them together, the result is zero.

For example in the fifth row, (+1-5+10-10+5+1=0). There are also interesting patterns if we consider if the values are odd or even as shown in Figure 1 below. Figure 1: Odd-Even Pascal’s triangle. The black and white circles represent odd and even numbers respectively. The figure also shows that some rows contain exclusively odd numbers, and each of the numbers is one less than a perfect power of 2 such as 1, 3, 7, 15, and 31. The explanation to this is that since the addition of prevailing even and corresponding odd number, which give an even number and at the edges they are against an odd number, the width will slowly decrease to a point (Cullinane, pp. 145-178).

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