Lacsap's Fractions Portfolio

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Portfolio #1
This portfolio surrounds the mathematical ideals of the LACSAP’s Fractions, and creating the task of answering certain questions about a specific symmetrical pattern. Through a work entirely of my own and without any unauthorized outside assistance, I answers all of the questions in this portfolio along with showcasing all my work, aided by the use of technology and patterns discovered by me.

The symmetrical pattern provided possesses only 5 vertical rows, with number of elements r increasing by 1 per new row created, and with r=0 representing the first element on each. However, one of the tasks requires finding the 6th row and its elements through patterns found in the pattern itself. Through a close analysis of the symmetrical pattern, which resembles Pascal’s Triangle in many ways, I found a relation between the numerators of the 1st element of every row, with r=1. As the first element of row 1 equals 1 (which can also be written as 11), the first element of row 2 equals 32. Just through that it can be seen the difference between numerators equaling 2. And as the first element in the third row is 64, the difference between numerators of second and third rows equals 3. And as I continued to analyze the numerators on the any elements of each of the following two rows, I came to the conclusion there was a pattern between their numerators and their row numbers. Therefore, the numerator of any element in any row will result from next numerator=previous numerator+(previous difference+1) equation. To be more mathematical, I developed the equation of numerator(row n)=n2+n2 , with n equaling the row number. To validate my general statement for finding numerators for rows, I tested it for finding the 6th row’s numerator, common to all of its elements. I calculated numerator6th row=n2+n2= 62+62=36+62=422=21, and through re-checking the patterns I’d found earlier and applying it to that row I came to the same results. However, the task was not fully...
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