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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. 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

Middle

denominator:

= (7) × (7+1) - 1(7-1)

= 28 – 6

= 22

2nd denominator:

= (7) × (7+1) - 2(7-2)

= 28 – 10

= 18

3rd denominator:

= (7) × (7+3) - 1(7-3)

= 28 – 12

= 16

4th denominator:

= (7) × (7+1) - 4(7-4)

= 28 – 12

= 16

5th denominator:

= (7) × (7+1) - 5(7-5)

= 28 – 10

= 18

6th denominator

= (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 1

1  1

1   1

1    1

1     1

1      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 ( + ) n2 + n – r (n-r)

‘n’ is the row number

‘r’ is the element

Validity:

Conclusion

r ( + )

‘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 1

1  1

1   1

1    1

1     1

1      1

1       1

1        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: n2 + n – r (n-r)

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

n ( + ) n2 + n – r (n-r)

Is our general statement.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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