(2+1)C2
(3)C2
-When n=5
(5+1)C2
(6)C2
15
Caption: The row numbers above are randomly selected within a range of 0≤x≤5.
Therefore, the numerator of 6th row can be found by,
(6+1)C2
(7)C2
x = 21 [Eq. 2]
and the numerator of 7th row also can be found by,
(7+1)C2
(8)C2
x = 28 [Eq. 3]
How to find denominator
Figure 3: The pattern showing the difference of denominator and numerator for each fraction. The first element and the last element are cut off since it is known that all of them are to be 1. However, only first row is not cut off.
Table 1: The table showing the relationship between row number and difference of numerator and denominator for each 2nd element
The difference of numerator and denominator increases by one. Moreover, it is clear that the difference between row number and difference of numerator and denominator is 1. Thus, the difference can be stated as (n-1). Therefore, the denominator of the 1st element can be shown as,
[Eq. 4]
Table 2: The table showing the relationship between row number and difference of numerator and denominator for each 1st element
The difference of numerator and denominator in 1st row is not applicable since there is no 2nd element in the 1st row. The difference of numerator and denominator increases by two. Thus, the difference can be stated as 2(n-2). Thus, the relationship between denominator and numerator can be shown as,
[Eq.5]
where n is row number.
From these data, it can be detected that there is a pattern that the number used in those equation is same as the element number. Therefore, the denominator can be stated as,
[Eq.6]
where n represents the row number and r represents the element number.
Sample Calculation
-When n = 4, and r = 3
Thus, the denominator in 6th row can be solved as,
1st element
2nd element
13
3rd element
12
4th element
13
5th element
Therefore, the pattern in 6th row is
Also, the denominator in 7th row can be solved as,
1st element
2nd element
18
3rd element
16
4th element
16
5th element
6th element
Hence, the pattern in 7th row is,
Conclusion
Therefore, the general statement of the rth element in nth row can be shows as,
[Eq.7]
where r is element number,
However, there are several limitations for this equation. First, number 1 located in both side of the given pattern should be cut out when the numerator is calculated. Thus, the second element of each row is counted as “the first element.” Second, n in general statement of numerator must be greater than 0. Third, the very first row of the given pattern is counted as the 1st row.
