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

Extracts from this document...

Introduction

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.  image30.jpg

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

...read more.

Middle

 denominator:

= image00.png

(7) × (7+1) - 1(7-1)

= 28 – 6

= 22

2nd denominator:

= image00.png

(7) × (7+1) - 2(7-2)

= 28 – 10

= 18

3rd denominator:

= image00.png

(7) × (7+3) - 1(7-3)

= 28 – 12

= 16

4th denominator:

= image00.png

(7) × (7+1) - 4(7-4)

= 28 – 12

= 16

 5th denominator:

= image00.png

(7) × (7+1) - 5(7-5)

= 28 – 10

= 18

6th denominator

= image00.png

(7) × (7+1) - 6(7-6)

= 28 – 6

= 22

From our previous investigation, we have already learned that the numerator of row seven will consistently be 28.

1          1

1          image01.png

         1

1          image02.png

image02.png

          1

1          image03.png

image04.png

image03.png

          1

1          image05.png

image06.png

image06.png

image05.png

          1

1          image10.png

image11.png

image12.png

image11.png

image10.png

           1  

1          image14.png

image15.png

image16.png

image16.png

image15.png

image14.png

1

General Statement:

To find the general statement for En (r), where the (r + 1)th element in the in the nth row. This is starting with r = 0 by combining the equation for the numerator and the denominator. The equation is:

n (image08.png

+image00.png

)

image00.png

n2 + image00.png

n – r (n-r)

‘n’ is the row number

‘r’ is the element

Validity:

...read more.

Conclusion

r (image29.png

+image00.png

)

‘r’ is the row number and by solving the equation, we can find the numerator.

When we observe the denominator in each row, we can see that there is a sequence. Starting in row 3, the denominators begin to change.

1          1

1          image01.png

         1

1          image02.png

image02.png

          1

1          image03.png

image04.png

image03.png

          1

1          image05.png

image06.png

image06.png

image05.png

          1

1          image10.png

image11.png

image12.png

image11.png

image10.png

           1  

1          image14.png

image15.png

image16.png

image16.png

image15.png

image14.png

1

1          image21.png

image22.png

image23.png

image24.png

image23.png

image22.png

image21.png

1

1          image25.png

image26.png

image27.png

image28.png

image28.png

image27.png

image26.png

image25.png

1

We can observe that the left side of the sequence is the same as the right side of the sequence. The element number matches the difference between the denominator and the numerator.

The equation used to fine the denominator is:

image00.png

n2 + image00.png

n – r (n-r)

Through the combination of these two equations we can come to the conclusion that,

n (image08.png

+image00.png

)

image00.png

n2 + image00.png

n – r (n-r)

Is our general statement.

...read more.

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