In Lacsap's Fractions, when looking for a general pattern for the numerator, it can be noted that it does not increase linearly but exponentially. Numerators are 3,6,10, and 15, each preceding numerator added by one plus the row number. Using this general statement it can be concluded that the numerator in the 6th row is 21 (15+6), and 28 for the 7th.
Generating a Statement for the Numerator:
To generate an equation for the numerator of the fraction, the fraction data must be organized and graphed. The table below shows the relationship between the row number and numerator being relative to an exponential function as the sequence goes on. N(n+1)-Nn represents the equation for the graph that increases more evenly as the sequence advances.
Using excel to graph the points and loggerpro to generate an equation, the general statement for finding the numerator N=0.5n2+0.5n, n having to be greater than 0. To check the validity of the equation sample equations were used: Sample Equation:
5th Row: N=0.5(5)2+0.5(5)=15
The first pattern that could be recognized is that the difference between the numerators of the ensuing rows is 1 more than the change between the previous numerator of the two consecutive rows. The formula that represents the pattern of how to find the numerator is N(n+1)-N(n)=N(n)-N(n-1)+1. Using this method, the 6th and 7th rows can be found:
This is only a supplement to the equation found in the graph above (N=0.5n2+0.5n). This pattern only tests the validity of the equation derived from the table because of both methods concluding to the same value.
Generating a Statement for the Denominator:
To examine the denominators in Lascap's Fractions, the values for the 6th row and their corresponding elements were put onto a table, and ultimately a graph. Showing a pattern, it...
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