Figure 2: The difference of the numerators between each row of Lascap’s fractions
In this case the difference of the denominators is 2, 3, 4, and 5 going down the rows. Based on these numbers only, the pattern is a sum notation. Which we can calculate, the common difference increases by each row and the elements have no effect on the difference so they are not included in the equation.
Table 1: The numerators of the first five rows of Lascap’s fractions
From the table this equation can be produced:
(1+…+n)
=
By using this equation one can calculate the numerator of the sixth row with the n, the number or rows.
n=6
=
=21
Graphical representation of correlation of the row number, n, and the numerator
Table 2: Numerators of row’s #1-20
Graph 1:
The slope of the line in the graph best corresponds to a polynomial function. As the equation of the line is a sum notation.
Finding the sixth and seventh rows using patterns:
Figure 3: Differences of each row’s denominators in Lascap’s fractions
The pattern found used to calculate the sixth and seventh rows is shown in figure 3. The numerator is calculated the same way as before with the equation,
. The denominator is can also be calculated with a similar formula as the numerator. Going down diagonally to the right the number that is being added is increased by one each row. An example is in row 1,element 2 the denominator is increased by 1 as it goes to row 2,element 2. Then in row 2, element 2 the denominator is increased by 2 as it goes to row 3, element 2. This pattern keeps on going up to row 5, element 2. Since there is an increase of 4 from row 4, element 2 to row 5, element 2, it is logical to assume there will be an increase of 5 from row 5, element 2 to row 6, element 2, making the denominator 16.
Sixth row: 1,
,
,
,
,
, 1
Seventh row: 1 ,
,
,
,
,
,
, 1
The general statement:
Let En(r) be the (r+1)th element in the nth row, starting with r=0
Find the general statement for En(r):
First break down the first 5 rows into their simplest form
From this I deduced a pattern for the fifth row
=
=
This makes the pattern for every element in the fifth row
=
The pattern can be further changed to
=
=
This led me to the general statement of:
Testing the validity of the general statement by finding additional rows
Table 3: The
Sample calculation: Row 8, element 2, where n=8 and r=1
=
=
=
Conclusion:
After validating, the equation for Lascap’s fraction is:
Where n is the row number and r is the (r+1)th element
There are however, limitations to the equation. The n and r cannot be a fraction, an imaginary number, or negative number. This is because n represents the nth row and r represents the (r+1)th element and neither the row number nor the element number can be a fraction unless the fraction equals to one because the row number or the element can only be a whole number. The same applies to imaginary numbers or negative numbers. It is still possible to substitute a fraction or a negative number into the general statement and get an answer however it will not mean anything. Zero is only used to substitute for r in the general equation, but it can also be substituted for n. This substitution changes the general statement from
to
, which still allows the formula to operate the same way because essentially all it does is add a 0 to all the elements.
