LACSAP’S FRACTIONS

10/11/2012

Tracy Braganza IB2T

In mathematics, Lacsap’s fractions are based upon Pascal's triangle. In this portfolio, the aim that was given was to consider a set of numbers that are presented in a symmetrical pattern, deduce a general statement and also to determine the limitations of the general statement that have been found. The answers in this portfolio will be attained with the help of a GDC calculator (GDC – TI84 Plus Silver Edition).

Investigation:

Considering the five rows of numbers shown below

1 1

1 32 1

1 64 64 1

1 107 106 107 1

1 1511 159 159 1511 1

In order to find the numerator of the sixth row we have been given the numerators of the five rows shown above. Excluding the first and last numerators of each row, thus we will consider the numerators 1, 3,6,10, and 15.

Figure 1:

This figure will portray the relationship between the first row and second row, second row and third row, third row and fourth row, and fourth row and fifth row. Through the figures obtained we will be able to justify the relationship between the fifth row and sixth row.

Row| Numerator| 1st differences | 2nd differences|

1| 1| 2| 1|

2| 3| | |

| | 3| |

3| 6| | 1|

| | 4| |

4| 10| | 1|

| | 5| |

5| 15| | |

Thus with the help of the figure above it is evident to say that the difference between the numerator in row 1 and row 2 is 2, row 2 and row 3 is 3, row 3 and 4 is 4 and row 4 and 5 is 5. The second difference for each row number is 1 showing that the equation for the numerator is a quadratic sequence. In order to acquire the numerator of the sixth row I will find the equation of the sequence by using the general form of a quadratic equation formula, y = ax2 + bx + c, where y = the numerator and x = the row number. In order to calculate the general statement for the values a and b I will use simultaneous equations through substitution method. I will use values from the second row x=3 and y=2

y = ax2 + bx + c

3=a (2)2+ b (2) +0

3= 4a + 2b

Then make b the subject

b = 3-4a2

b= 1.5 – 2a

Then using the third row x=3 and y= 6

y = ax2 + bx + c

6= a (3)2+ b (3) +0

6= 9a+ 3b

The next step is to substitute b into the equation of the third row 6=9a+ 3(1.5 – 2a)

6= 9a+ 4.5 – 6a

6=3a+ 4.5

6 – 4.5= 3a

0.5=a

Whereas:

b=1.5 – 2(0.5)

b=1.5 – 1

b=0.5

Therefore: a=0.5 and b=0.5

So the equation for the numerator is 0.5n2 + 0.5n, where n = the row number. With the help of this equation, the numerator of the sixth row is: 0.5n2 + 0.5n

0.5(6)2 + 0.5 (6)

0.5(36) + 3

18 + 3 = 21

Thus the numerator of the sixth row is 21.

The equation 0.5n2 + 0.5n can be simplified to represent

Nn= 0.5n2+ 0.5n

N= numerator

n= row number

Nn= n(n+1)2

In order to justify the equation of the numerator the figure below shows the relationship between the row number n and the numerator N in each row.

ROW NUMBER| NUMERATOR|

4| 4(4+1)2= 10|

5| 5(5+1)2= 15|

6| 6(6+1)2= 21|

7| 7(7+1)2= 28|

9| 9(9+1)2= 45|

15| 15(15+1)2=120|

20| 20(20+1)2= 210|

Figure 2:

The figure above shows the relationship between the numerator and row number. The row number is on the x-axis and the numerator is on the y-axis. It is evident that as the number of rows increases so does the value of the numerator. Since the trend line of the graph is not straight it shows that the equation of the line is not linear and thus the line has been plotted with the equation 0.5x2 + 0.5x. This proves that the equation of the numerator is correct. In order to calculate the sixth and seventh row, the equation of the denominator of each row has to be made known. When looking at the Lacsap’s fraction again:...