LACSAP Fractions. The aim of this portfolio is to discover an equation which suits the pattern of Lacsaps fraction using technology

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4/27/2012


LACSAP’S FRACTIONS

Lacsap is backward for Pascal, therefore we can predict that the Pascal’s triangle can be applied in answering the following questions.

The aim of this portfolio is to discover an equation which suits the pattern of Lacsap’s fraction using technology. The equation will have to consistently determine and follow the correct pattern for the numerator and denominator of each row.  

Finding the Numerator:

The numerators in Lacsap’s fraction pattern follow the pattern of the third element of r=2 in Pascal’s triangle. Thus, the numbers with a circle in figure 1 are the same as the numerators in Lacsap’s fractions.          

Consequently, the equation used to represent the numerator can be:

Numerator =  

 n × (n+1)

‘n’ is the representation of the row number.

‘x’ is the unknown numerator of the sixth row

To find the numerator of the sixth row we would replace ‘n’ with 6.

(6) × (6+1) = x

3 × 7 = 21

x = 21

1          1

1          

          1

1          

          

          1

1          

         

         

          1

1          

         

         

         

          1

Plotting the Relation:

The row number, which has been plotted up to the ninth row, is represented by the x-axis and the numerator by the y-axis. There is a parabolic curve to the plot. This means the numerator is increasing at a higher ratio than the row number. The sequences of the numerators are 3, 6, 10, and 15. When analyzed, we can see that the numerators are increasing one more than the previous term. 3 is increased by three to equal 6. 6 is increased by 4 to equal 10 and so forth.  From this pattern, we can formulate an equation using the row number as a variable to find the numerator.

We use the row number because we can observe that to get from row number, 2, to the numerator of the row, 3, we must multiply the row number by

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 .

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                     ...

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