.
2 ×
= 3
By using row 3 we can find 6, the numerator of row 3, if we multiply by 2.
3 × 2 = 6
To find the numerator of 10 of row number 4, we must multiply by
.
4 ×
= 10
From this, we can observe that to find the numerator using the row number, we must multiply the row number by a certain constant plus
more than the previous row. The equation derived from this information would be:
EQUATION n (
+
)
Finding the Numerator for the Sixth Row:
The numerator of the sixth row can be found by using the equation shown above and by replacing ‘r’ with 6.
6(
+
) = 21
Finding the Denominator:
To find the denominator, we will first have to find the difference between the current and previous denominators. This is done by subtracting the denominator from the numerator.
Take the fraction
. 3 is the numerator and 2 is the denominator. Subtract 2 from 3 to get the difference 1. Now repeat this process for the other fractions.
The general statement needed to find the denominator of the Lacsap’s fractions is:
D(r) =
n × (n+1) - r (n-r)
‘D’ represents the denominator, ‘n’ is the row number, and ‘r’ is for representing the element number
To Find the Sixth Row:
When we look for the sixth row, we can make use of the previous equations.
Numerator:
= 6(
+
)
= 21
1st denominator:
=
(6) × (6+1) - 1(6-1)
= 21 – 5
= 16
2nd denominator:
=
(6) × (6+1) - 2(6-2)
= 21 – 8
= 13
3rd denominator:
=
(6) × (6+1) - 3(6-3)
=21 – 9
=12
4th denominator:
=
(6) × (6+1) - 4(6-4)
=21 – 8
= 13
5th denominator:
=
(6) × (6+1) - 5(6-5)
= 21 – 5
= 16
From our previous calculations, we have learned that for the sixth row, the numerator will always be 21 for the fractions. With the sixth row, Lacsap’s fractions will look like this.
1 1
1
1
1
1
1
1
1
1
1
1
To Find the Seventh Row:
When finding the seventh row, we can use the same equation we used to find the sixth row.
Numerator:
= 7(
+
)
= 28
1st denominator:
=
(7) × (7+1) - 1(7-1)
= 28 – 6
= 22
2nd denominator:
=
(7) × (7+1) - 2(7-2)
= 28 – 10
= 18
3rd denominator:
=
(7) × (7+3) - 1(7-3)
= 28 – 12
= 16
4th denominator:
=
(7) × (7+1) - 4(7-4)
= 28 – 12
= 16
5th denominator:
=
(7) × (7+1) - 5(7-5)
= 28 – 10
= 18
6th denominator
=
(7) × (7+1) - 6(7-6)
= 28 – 6
= 22
From our previous investigation, we have already learned that the numerator of row seven will consistently be 28.
1 1
1
1
1
1
1
1
1
1
1
1
1
1
General Statement:
To find the general statement for En (r), where the (r + 1)th element in the in the nth row. This is starting with r = 0 by combining the equation for the numerator and the denominator. The equation is:
n (
+
)
n2 +
n – r (n-r)
‘n’ is the row number
‘r’ is the element
Validity:
To prove that this equation is valid, we will use it to find rows eight and nine.
To Find the Eighth Row:
Numerator:
= 8(
+
)
= 36
1st denominator:
=
(8) × (8+1) - 1(8-1)
= 36 – 7
= 29
2nd denominator:
=
(8) × (8+2) - 1(8-2)
= 36 – 12
= 24
3rd denominator:
=
(8) × (8+1) - 3(8-3)
= 36 – 15
= 21
4th denominator:
=
(8) × (8+1) - 4(8-4)
= 36 – 16
= 20
5th denominator:
=
(8) × (8+1) - 5(8-5)
= 36 – 15
= 21
6th denominator:
=
(8) × (8+1) - 6(8-6)
= 36 – 12
= 24
7th denominator:
=
(8) × (8+1) - 7(8-7)
= 36 – 7
= 29
In the graph above, we can see that the parabola passes through all of the valid points. The ‘1’ is made into
to be easily checked. We can now see that the calculations made using the general statement are correct.
To Find the Ninth Row:
Numerator:
= 9(
+
)
= 45
1st denominator:
=
(9) × (9+1) - 1(9-1)
= 45 – 8
= 37
2nd denominator:
=
(9) × (9+1) - 2(9-2)
= 45 – 14
= 31
3rd denominator:
=
(9) × (9+1) - 3(9-3)
= 45 – 18
= 27
4th denominator:
=
(9) × (9+1) - 4(9-4)
= 45 – 20
= 25
5th denominator:
=
(9) × (9+1) - 5(9-5)
= 45 – 20
= 25
6th denominator:
=
(9) × (9+1) - 6(9-6)
= 45 – 18
= 27
7th denominator:
=
(9) × (9+1) - 7(9-7)
= 45 – 14
= 31
8th denominator:
=
(9) × (9+1) - 8(9-8)
= 45 – 8
= 37
This graph shows us that the parabola goes through all of the valid points for the denominators of the ninth row. 1 has been changed to
so that the general statement could be checked for validity.
The Scope/ Limitations of the General Statement:
The general statement works for many of the elements, but it does have its limitations. The 1 cannot be negative because the general statement would not work if the parabola is facing down, or in other words negative. The general statement can only be valid if r ≠ 0 and r ≠ n because the first and last number of each row is 1.
Conclusion:
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
To find the numerator we used the equation:
r (
+
)
‘r’ is the row number and by solving the equation, we can find the numerator.
When we observe the denominator in each row, we can see that there is a sequence. Starting in row 3, the denominators begin to change.
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
We can observe that the left side of the sequence is the same as the right side of the sequence. The element number matches the difference between the denominator and the numerator.
The equation used to fine the denominator is:
n2 +
n – r (n-r)
Through the combination of these two equations we can come to the conclusion that,
n (
+
)
n2 +
n – r (n-r)
Is our general statement.