= 0.5(25) + 2.5
= 12.5 + 2.5
Numerator = 15 when n is 5 the numerator is 15
Find numerator(N) value when n is equal to 6
N = 0.5n2 + 0.5n
= 0.5(6)2 + 0.5(6)
= 0.5(36) + 3
= 18 + 3
N = 21
Find numerator value when n is equal to 7
N = 0.5n2 + 0.5n
= 0.5(7)2 + 0.5(7)
= 0.5(49) + 4.5
= 24.5 + 3.5
= 28
To check my N value I will continue the pattern into the seventh row
1 + 2 =3…(n=3)....3 + 3 = 6…(n=4)…..6 + 4 = 10…(n=5)…..10 + 5 = 15…(n=6)…..15 + 6 = 21…
…(n=7)...21 + 7 = 28
Correct value for N
After finding the numerator values of the sixth and seventh row my next step will be to find the demonimator values. As the numerator values have formed a pattern in diagonal lines, so did the denominator. However a difference I noticed was that the numerator values were consistent with each row but when it came to the denominator there were different values within each row. Despite the difference one thing that was similar between the numerator and denominator would be the way each term is increasing.
Now that I have figured out a pattern for both the denominator and the numerator, the next step I will take in my investigation will be to find a general statement which represents the denominator. Where r represents the different elements or terms in each row, starting with r = 0 and n represents the row number.
In order to find a general statemnet for the denominator I will be considering the relationship between the numerator and denominator. I began to notice a pattern formulating between the r value and the difference between the numerator and denominator.
The table below shows the relationship between the row number(n) and the difference between the numerator and denominator when the element or r is equal to 1.
The difference when r is equal to 1 can be stated as (n -1)
The difference in the first row is not applicable because there is no second element in the first row. The differece increase by 2 each time and therefore the difference can be stated by 2(n-2).
Again, since there is no third element in rows 1 and 2 the difference is not applicable. The difference this time increases by 3 each time and therefore the difference can be stated as 3(n-3).
Again, since there is no fourth element in rows 1,2 and 3 the difference is not applicable. The differecen this time increases by 4 each time and therefore the difference can be stated as 4(n-4).
For this table rows 1,2,3 and four aren’t applicable because there is no fifth element in those rows. The difference this time increases by 5 each time and therefore the difference can be stated as 5(n-5).
Looking at these tables it becomes obvious that the difference between numerator and denominator increases by the r value. Thus I was able to conclude that the general statement much be something along the lines of 0.5n2 + 0.5n – (difference of Numerator and Denominator). Looking at the data I have gathered above I have come to the conclusion that the general statement must be…..
Denominator = (0.5n2 + 0.5n) – r(n – r)
Now that I have come up with my general statement I will test its validity to make sure it works.
Test validity for when denominator = 7, n = 4 and r=3
Denominator = (0.5n2 + 0.5n) – r(n – r)
7 = (0.5(4)2 + 0.5(4)) – 3(4 – 3)
7 = (0.5(16) + 2) – 3(1)
7 = (8 + 2) – 3
7 = 10 – 3
7 = 7
Test validity for when denominator = 18, n = 7, r = 5
Denominator = (0.5n2 + 0.5n) – r(n – r)
18 = (0.5(7)2 + 0.5(7)) – 5(7 – 5)
18 = (0.5(49) + 3.5) – 5(2)
18 = (24.5 + 3.5) – 10
18 = 28 – 10
18 = 18
Now that I have arrived at a general statement for both the numerator and denominator individually, the next step in my investigation would be to find the general statement which represents both the numerator and denominator of an element simultaneously. By letting the element be represented by En(r) where n represents the row number and r represents the element in each row. By combining the general statements for both the numerator and denominator the general statement I have come up with is……
En(r) =
Now that I have come up with the general statement I will test its validity to make sure my statement is valid.
Test validity for when En(r) =
and n = 3 and r = 1
En(r) =
=
=
=
=
Test validity for when n = 10 and r = 6 however since I have not extended my pattern into the 10th row I will extend my pattern in the 10th row for the 6th element in order to figure what the En(r) value will be. Then I will use my general statement to test its validity.
Numerator value = 55
Denominator value = 31
∴ En(r) =
Test general statement validity:
En(r) =
=
=
=
=
Test validity for when n = 12 and r = 3 however since I have not extended my pattern into the 10th row I will extend my pattern in the 12th row for the 3th element in order to figure what the En(r) value will be. Then I will use my general statement to test its validity.
Numerator = 78
Denominator = 51
∴ En(r) =
Test general statement validity:
En(r) =
=
=
=
=
After testing the validity of my general statement I have come to the conclusion that my statement is valid for all the number I have tested. However there are some limitations to my general statement, regarding the variable I am using. Firstly n must be a whole number because there needs to be an entire row to complete the pattern, you cannot have part of the row, n must also be greater than zero because there can’t be a negative row in the pattern. The same limitations apply to r, which must be greater than zero and a whole number because there cannot be a negative or partial element in the pattern.
I was able to arrive at my general statement by considering patterns for both the denominator and numerator separately. This helped me focus on smaller parts of the final general statement and enabled me to notice all the different patterns that were forming within the lacsap fractions. By only focusing on one part of the fraction, coming up with a general statement for the numerator and denominator became very simple and straight forward. With those two general statements, I was later able to collaborate them together and arrive at a general statement for the entire fraction.
by
lil_nerd94 (student)
