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  • Level: GCSE
  • Subject: Maths
  • Word count: 1506

Patterns With Fractions Investigations

Extracts from this document...

Introduction

Ahzaz Chowdhury 11H

Fractions Coursework.

Mathematics Coursework.

Patterns With Fractions.

image00.jpg

Consider the sequence of fractions and the differences between the fractions:

Term (n)          1                    2                    3                    4                    5

image01.pngimage24.pngimage26.pngimage27.pngimage30.png

1st Difference           image23.pngimage02.pngimage25.pngimage10.png

2nd Difference                    image02.pngimage10.pngimage21.png

(For rest of differences, see page11)

Finding the starting fraction for the nth term:

image01.png, image24.png, image26.png, image27.png, image30.png     =     image04.png(The general formula)

Check if correct formula:

Term

(n)

Numerator

(n)

Denominator

(n + 1)

Final

Fraction

1

1

2

image01.png

2

2

3

image24.png

3

3

4

image26.png

4

4

5

image27.png

5

5

6

image30.png

(Check On Page 11)

Finding the nth term for the 1st difference:

In order to find out the nth term for the 1st differences, the requirement is to subtract the 2nd fraction from the 1st fraction (the smaller fraction from the bigger fraction).

image52.png  -  image53.png

=                 image54.png

=                image55.png

=                image05.png(The general formula for 1st difference)

Check if correct formula:

Term

(n)

Numerator

(1)

Denominator

(n + 1)(n+2)

Final

Fraction

1

1

(1+1)(1+2) =

6

image23.png

2

1

(2+1)(2+2) =

12

image02.png

3

1

(3+1)(3+2) =

20

image25.png

4

1

(4+1)(4+2) =

30

image10.png

5

1

(5+1)(5+2) =

42

image29.png

(Check On Page 11)

Finding the nth term for the 2nd difference:

In order to find out the nth term for the 2nd differences, the requirement is to subtract the 1st fraction from the 2nd fraction (the smaller fraction from the bigger fraction) of the 1st differences.

  -  image56.png

=                image57.png

=                image06.png(The general formula for 2nd difference)

Check if correct formula:

Term

(n)

Numerator

(2)

Denominator

(n + 1)(n+2)(n+3)

Fraction

Final

Fraction

1

2

(1+1)(1+2)(1+3) =

24

image20.png

image02.png

2

2

(2+1)(2+2)(2+3) =

60

image58.png

image10.png

3

2

(3+1)(3+2)(3+3) =

120

image59.png

image21.png

4

2

(4+1)(4+2)(4+3) =

210

image60.png

image17.png

5

2

(5+1)(5+2)(5+3) =

504

image61.png

image39.png

(Check On Page 11)

Finding the nth term for the 3rd difference:

...read more.

Middle

image37.png

(Check On Page 11)

Finding the nth term for the 4th difference:

In order to find out the nth term for the 4th differences, the requirement is to subtract the 1st fraction from the 2nd fraction (the smaller fraction from the bigger fraction) of the 3rd differences.

image07.png  -   image70.png

=        image71.png

=        image72.png

=         image08.png

(The general formula for 4th difference)

Check if correct formula:

Term

(n)

Numerator

(24)

Denominator

(n + 1)(n+2)(n+3)(n+4)(n+5)

Fraction

Final

Fraction

1

24

(1+1)(1+2)(1+3)(1+4)(1+5) =

720

image73.png

image10.png

2

24

(2+1)(2+2)(2+3)(2+4)(2+5) =

2520

image74.png

image75.png

3

24

(3+1)(3+2)(3+3)(3+4)(3+5) =

6720

image76.png

image31.png

4

24

(4+1)(4+2)(4+3)(4+4)(4+5) =

15120

image77.png

image35.png

5

24

(5+1)(5+2)(5+3)(5+4)(5+5) =

30240

image78.png

image40.png

(Check On Page 11)

Finding the nth term for the 5th difference:

In order to find out the nth term for the 5th differences, the requirement is to subtract the 1st fraction from the 2nd fraction (the smaller fraction from the bigger fraction) of the 4th differences.

image08.png   -   image79.png

=        image80.png

=        image81.png

=        image09.png

(The general formula for 5th difference)

Check if correct formula:

Term

(n)

Numerator

(120)

Denominator

(n+1)(n+2)(n+3)(n+4)(n+5)

(n+6)

Fraction

Final

Fraction

1

120

(1+1)(1+2)(1+3)(1+4)(1+5)

(1+6) =

5040

image82.png

image29.png

2

120

(2+1)(2+2)(2+3)(2+4)(2+5)

(2+6) =

20160

image33.png

3

120

(3+1)(3+2)(3+3)(3+4)(3+5)

(3+6) =

60480

image83.png

image37.png

4

120

(4+1)(4+2)(4+3)(4+4)(4+5)

(4+6) =

151200

image84.png

image40.png

5

120

(5+1)(5+2)(5+3)(5+4)(5+5)

(5+6) =

332640

image85.png

image86.png

(Check On Page 11)

In order to check if the formulas I have worked out are correct, it is best to justify one of the differences as justifying one of them proves that the rest of the formulas must be correct. In order make a fair generalisation; I am going to justify the 2nd difference (denominator) formula.

Justification of the 2nd difference:

The formula for the 2nd difference:

image06.png

In order to justify the formula, it is best to use the area of the formula that alters the most between formulas to formula. This is the case of the denominators in the formulas.

The formula for the denominator is [(n+1)(n+2)(n+3)].

The actual denominators for the second differences are:

Denominator

12

30

60

105

168

1st difference

18

30

45

63

2nd difference

12

15

18

3rd difference

3

3

As can be seen, in order to find a constant difference, the third difference for the denominators had to be found.

This therefore shows that the general formula is:

y = an3 + bn2 + cn + d

In the above formula, there are four unknowns; these have to be worked out in order to justify the formula.

Work out 1st unknown (the constant a)

In order to work out the constant a, the rule for the 3rd difference must be used.

a=  image23.png of  3                                            image03.pnga  =  image01.png

Work out 2nd, 3rd and 4th (the constants b, c and d)

Using the fact that a = image01.png, put the first constant in the formulay = an3 + bn2 + cn + d

Formula No

Term (n)

y = an3 + bn2 + cn + d

Final Formula

1

1

12 = 0.5 + b + c + d

b + c + d  = 11.5

2

2

30 = 4 + 4b + 2c + d

4b + 2c + d  = 26

3

3

60 = 13.5 + 9b + 3c + d

9b + 3c + d  = 46.5

...read more.

Conclusion

c5 c21">=      y = image01.pngn3 + 3n2 + 5.5n + 3

Term

(n)

y = image01.pngn3 + 3n2 + 5.5n + 3

Denominator

(y)

1

� + 3 + 5.5 + 3

12

2

4 + 12 + 11 + 3

30

3

13.5 + 27 + 16.5 + 3

60

4

32 + 48 + 22 + 3

105

5

62.5 + 75 + 27.5 + 3

168

* Note: The formula is correct. *

In order to transform y = image01.pngn3 + 3n2 + 5.5n + 3 so that there is no longer any decimals or fractions in it, I shall multiply the denominator formula by 2.

2 [image01.pngn3 + 3n2 + 5.5n + 3]

= n3 + 6n2 + 11n + 6

Check if above formula is correct:

(n + 1)(n + 2)(n + 3)

        = (n2 + 2n + n + 2)(n+3)

        = n3 + 2n2 + n2 + 2n + 3n2 + 6n + 3n + 6

        = n3 + 6n2 + 11n + 6

image03.pngFormula is correct ()

Finding the general term:

So far, I have made formulas to interrogate the desired fraction through inputting the term number. A general term would be much easier to use if I could enter the term number and also the difference number and be left with the fraction that is required.

Below are all of the different formulas for the different differences.

  • Starting Fractionimage04.png
  • 1st Difference        image05.png
  • 2nd Difference        image06.png
  • 3rd Difference        image07.png
  • 4th Differenceimage08.png
  • 5th Difference        image09.png

The general term for the fractions is:

image11.png

Check if correct formula:

Term

(n)

Difference

(x)

Numerator

(n! x!)

Denominator

(n + x + 1)!

Fraction

Final Fraction

6

4

720 * 24

(6 + 4 + 1)!

image12.png

image13.png

5

5

120 * 120

(5 + 5 + 1)!

image14.png

image15.png

4

2

24 * 2

(4 + 2 + 1)!

image16.png

image17.png

3

3

6 * 6

(3 + 3 + 1)!

image18.png

image19.png

2

1

2 * 1

(2 + 1 + 1)!

image20.png

image22.png

(Check on page 11)

Checking Sheet

Starting Fraction

1st Difference

2nd Difference

3rd Difference

4th Difference

5th Difference

image01.png

image23.png

image24.png

image02.png

image02.png

image25.png

image26.png

image10.png

image10.png

image25.png

image21.png

image27.png

image21.png

image17.png

image10.png

image28.png

image29.png

image30.png

image17.png

image31.png

image29.png

image32.png

image33.png

image34.png

image33.png

image35.png

image36.png

image37.png

image37.png

image38.png

image39.png

image40.png

image41.png

image42.png

image40.png

image43.png

image44.png

image45.png

image46.png

image47.png

image48.png

image49.png

image50.png

image51.png

        -  -

...read more.

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