Lacsap's Fractions : Internal Assessment

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040 Irwin Chan

Lacsap's Fractions : Internal Assessment

IB Math SL Type 1

Aim: In this task you will consider a set of numbers that are presented in a symmetrical pattern.

Figure 1 : Lacsap's Fractions

The first five rows of numbers are shown above. In order to find the numerator of the sixth row, I will use the numbers that go down the triangle diagonal, as shown from the highlighted fractions above. Hence the numerators are:

1        3        6        10        15

Figure 2 : Table showing relationship between n rows and numerator

The table above shows the relationship between row and numerator. The first difference between the numerator in row 1 and 2 was 2, between row 2 and 3 was 3, and so forth (2, 3, 4, 5). The second difference for each row number is 1, hence the equation for the numerator is a geometric sequence. Therefore, the find the equation of the sequence, the quadratic formula, y = ax2 + bx + c should be used, where y is the numerator and x is the row number.

To find this general statement for the numerator, I will calculate the values of a and b using simultaneous equations (substitution method):

Using the values from the table: x = 2 and y = 3 (second row)

Substitute into the quadratic formula (c is disregarded), and make b the subject:

3 = a (2)2 + b(2) + 0

3 = 4a + 2b

b = -2a + 1.5 

 Using the values from the third row : x = 3 and y = 6

6 = a (3)2 + b(3) + 0

6 = 9a + 3b

Substitute b = -2a + 1.5,

6 = 9a + 3(-2a + 1.5)

6 = 9a - 6a + 4.5

3a = 1.5

a = 0.5

Therefore,

b = -2(0.5) + 1.5

b = -1 + 1.5 = 0.5

We have come to a general statement for the nth term of the numerator:

Sn = 0.5n2 + 0.5n

Re-written:

Sn =

Another method involving the use of the TI-83 Plus Calculator can be used. By using the formula of 0.5n2 + 0.5n achieved from the method above, I entered it into the GDC.

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When the formula was entered into the GDC, the graph above was shown. This proves that it is a quadratic equation.

Therefore, the equation can be solved by using the Quadratic Regression 'QuadReg' function. The steps of this have been shown through a series of camera shots of the calculator.

1. Press the STAT button on the calculator, go to EDIT.

Once in EDIT, two empty columns will appear. Input the row number, n, in the first column, and ...

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