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• Level: GCSE
• Subject: Maths
• Word count: 1208

# Maths algerbra

Extracts from this document...

Introduction

Maths coursework

In  this coursework we were asked to find the differences in a simple fraction pattern. We had ;

 Number 1 2 3 4 5 6 7 Fraction 1/2 2/3 3/4 4/5 5/6 6/7 7/8

I will look at the differences between the fractions and search  for any general rules. I will apply my good algebraic skills to help find any rules.

I then had to find the differences by taking the fractions away from each other. To find my first difference I had to ;

D1 = difference one

Fraction         2nd-1st                 3rd-2nd                4th-3rd

D1                  1st                               2nd                              3rd

I had to take away 2/3 form  1/2 but the  denominators were not the same, so I found the lowest common multiple  from each and multiplied them top and bottom eg.

X2   2   -     1   X3

X2   3         2   X3

The lowest common  multiple is 6 so I multiplied  2/3 by 2 and 1/2 by 3. I then got ;

4/6 -  3/6 which then equalled 1/6, which was my 1st difference.

I done the same to 3/4 -  2/3             9/12 – 8/12 which equalled 1/12 and my second difference.

Middle

nd – 3rd                 3rd   – 4th

D2           1st                               2nd                               3rd

I then looked for the nth term in the second difference, I had to take the nth term away from the n + 1th  term and so I …

1          -           1

(n+1)(n+2)       (n+2)(n+3)

X (n + 3)             1      -           1                 X  (n + 1)

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

1X (n+3)                         -    1 X  (n + 1)

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

 X n 3 1 n 3 X n 1 1 n 1

(n + 3) – (n + 1)

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

which cancelled to

(n + 3)  - (n + 1)

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

2

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

I tested this by using n = 3

2

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

2

4 X 5 X 6

2

120

1

60

This is the 3rd difference in the second difference line.

I went on to find the 3rd difference.

 Number 1 2 3 4 5 n n+1 D2 1/12 1/30 1/60 1/105 2/(n+1)(n+2)(n+3) 2/(n+2)(n+3)(n+4) D3 1/20 1/60 1/60 1/140

I found this by taking the nth term away from the n+1th term ;

2                                             2

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

I had to make the denominators the same so I …

Conclusion

X1

X2

X3

X4

I predicted that the numorator would be 4 times more then the previous numerator because of the patten forming (24), I also predicted the denominator be times by one more then the number difference. To check this I had to test it I put n = 1

??        ??        12        24

2 X 3 X 4 X 5 X 6                         3360        3360        720

1

30

this 1makes my prediction right

I went on to find dk I predicted it to be

1 X 2 X 3 X 4… X k

(n+1)(n+2)(n+3)(n+4)…(n+1+k)

I proved this by ;

Numerator

 Difference D  1 D  2 D  3 D  4 D  k fraction 1 1X  2 1X2X 3 1X2X3X 4 1X2X3X 4…X k numerator 1 2 6 24 1X2X3X4..Xk

The difference number was the last consecutive number multiplied to get the numerator

## Denominator

 Difference D  1 D  2 D  3 D  k Actual fraction (n+1)(n+2) (n+1)(n+2)(n+3) (n+1)(n+2)(n+3)(n+4) (n+1)(n+2)(n+3)(n+4)…(n+K+1) explanation (n+1)(n+1+1) (n+1)(n+2)(n+2+1) (n+1)(n+2)(n+3)(n+3+1) (n+1)(n+2)(n+3)(n+4)(n+k+1)

This is how I explain the denominator of Dk.

Daisy Driscoll

10BA

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