- Level: International Baccalaureate
- Subject: Maths
- Word count: 2157
LACSAP FRACTIONS - I will begin my investigation by continuing the pattern and finding the numerator of the sixth row.
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Introduction
LACSAP’S FRACTIONS |
The purpose of my investigation will be to achieve the general statement for a set of numbers that are presented in a symmetrical pattern.
Looking at the pattern below I will begin my investigation by continuing the pattern and finding the numerator of the sixth row.
Observing this pattern the first thing I noticed was the value of the numerator is consistent in each row and I began to notice a pattern forming in diagonal lines. I noticed that the numerator would increase by a value that would increase by 1 in each row. For example looking at the example below:
Therefore I can conclude that the numerator value for the sixth row will be 21.
After continueing the pattern into the sixth row, to further my investigation I will write a general statement which will present the numerator. In order to find the general statement firstly I will plot the relation betwen the row number (n) and the numerator value in each row.
After considering the graph and the pattern in which the numerator value was increasing I came to the realization that the pattern was an arithmetic sequence. Therefore I was able to arrive at the general statement by simplifying the formula for an arithmetic sequence. Where U1 is equal to 1 and d is equal to 1.
Sn =
( 2u1 + (n – 1)d)
Middle
N/A
3
0
4
3
5
6
6
9
7
12
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).
Row number (r = 4) | Difference of Numerator and Denominator |
1 | N/A |
2 | N/A |
3 | N/A |
4 | 0 |
5 | 4 |
6 | 8 |
7 | 12 |
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).
Row number (r = 5) | Difference of Numerator and Denominator |
1 | N/A |
2 | N/A |
3 | N/A |
4 | N/A |
5 | 0 |
6 | 5 |
7 | 10 |
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.
Row Number (n) | Numerator |
2 | 1+2 = 3 |
3 | 3+3 = 6 |
4 | 6+4 = 10 |
5 | 10+5 = 15 |
6 | 15+6 = 21 |
7 | 21+7 = 28 |
8 | 28+8 = 36 |
9 | 36+9 = 45 |
10 | 45+10 = 55 |
Conclusion
Row Number (n) | Numerator |
2 | 1+2 = 3 |
3 | 3+3 = 6 |
4 | 6+4 = 10 |
5 | 10+5 = 15 |
6 | 15+6 = 21 |
7 | 21+7 = 28 |
8 | 28+8 = 36 |
9 | 36+9 = 45 |
10 | 45+10 = 55 |
11 | 55+11 = 66 |
12 | 66+12 = 78 |
Numerator = 78
Row number (r = 3) | Denominator |
1 | N/A |
2 | N/A |
3 | 6 |
4 | 6+1 = 7 |
5 | 7+2 = 9 |
6 | 9+3 = 12 |
7 | 12+4 = 16 |
8 | 16+5 = 21 |
9 | 21+6 = 27 |
10 | 27+7 = 34 |
11 | 34+8 = 42 |
12 | 42+9 = 51 |
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.
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