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
Patterns Recognized:
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:
6th:
N(5+1)-N(5)=N(5)-N(4)+1
N(6)-15=15-10+1
N(6)=15+6
N(6)= 21
7th:
N(6-1)-N(6)=N(6)-N(5)+1
N(7)-21=21-15+1
N(7)=42-15+1
N(7)= 28
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 was concluded that the denominator could be found with a general equation of D=r2-nr+r0. This was found through loggerpro. The element in the row was represented by r, and r0 being the first denominator value in the row. Lastly n represents the row number.
To test the equation, the 5th and 6th row denominators were put in, both finding the second denominator values.
5th: D=22-(5)(2)+15=11
6th: D=22-(6)(2)+21=13
After knowing how to determine both denominator and numerators, the Lacsap's Fraction triangle could be filled out to the 7th row.
General Statement Overall:
After determining equations for both numerator and denominator and following patterns, all Lacsap's Fractions could be found with a general statement of:
En(r)=
=
Although proven for each separate category of numerator and denominator, the general statement can be concluded valid with these tests.
Finding X row:
5th row 4th Fraction: E5(4)=
=
8th row 3rd Fraction: E8(4)=
=
Limits and Conditions:
Other than the Nth row having to be greater than 0, there is no limit to this equation as it can find any fraction within Lacsap's Fraction Triangle.