Patterns With Fractions Investigations
Extracts from this document...
Introduction
Ahzaz Chowdhury 11H
Fractions Coursework.
Mathematics Coursework.
Patterns With Fractions.
Consider the sequence of fractions and the differences between the fractions:
Term (n) 1 2 3 4 5
1st Difference
2nd Difference
(For rest of differences, see page11)
Finding the starting fraction for the nth term:
, , , , = (The general formula)
Check if correct formula:
Term (n)  Numerator (n)  Denominator (n + 1)  Final Fraction  
1  1  2  ✓  
2  2  3  ✓  
3  3  4  ✓  
4  4  5  ✓  
5  5  6  ✓ 
(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).

=
=
= (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  ✓  
2  1  (2+1)(2+2) = 12  ✓  
3  1  (3+1)(3+2) = 20  ✓  
4  1  (4+1)(4+2) = 30  ✓  
5  1  (5+1)(5+2) = 42  ✓ 
(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.

=
= (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  ✓  
2  2  (2+1)(2+2)(2+3) = 60  ✓  
3  2  (3+1)(3+2)(3+3) = 120  ✓  
4  2  (4+1)(4+2)(4+3) = 210  ✓  
5  2  (5+1)(5+2)(5+3) = 504  ✓ 
(Check On Page 11)
Finding the nth term for the 3rd difference:
Middle
✓
(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.

=
=
=
(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  ✓  
2  24  (2+1)(2+2)(2+3)(2+4)(2+5) = 2520  ✓  
3  24  (3+1)(3+2)(3+3)(3+4)(3+5) = 6720  ✓  
4  24  (4+1)(4+2)(4+3)(4+4)(4+5) = 15120  ✓  
5  24  (5+1)(5+2)(5+3)(5+4)(5+5) = 30240  ✓ 
(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.

=
=
=
(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  ✓  
2  120  (2+1)(2+2)(2+3)(2+4)(2+5) (2+6) = 20160  ✓  
3  120  (3+1)(3+2)(3+3)(3+4)(3+5) (3+6) = 60480  ✓  
4  120  (4+1)(4+2)(4+3)(4+4)(4+5) (4+6) = 151200  ✓  
5  120  (5+1)(5+2)(5+3)(5+4)(5+5) (5+6) = 332640  ✓ 
(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:
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= of 3 a =
Work out 2nd, 3rd and 4th (the constants b, c and d)
Using the fact that a = , 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 
Conclusion
Term (n)  y = n3 + 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 = n3 + 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 [n3 + 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
Formula 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 Fraction
 1st Difference
 2nd Difference
 3rd Difference
 4th Difference
 5th Difference
The general term for the fractions is:
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)!  
5  5  120 * 120  (5 + 5 + 1)!  
4  2  24 * 2  (4 + 2 + 1)!  
3  3  6 * 6  (3 + 3 + 1)!  
2  1  2 * 1  (2 + 1 + 1)! 
(Check on page 11)
Checking Sheet
Starting Fraction  1st Difference  2nd Difference  3rd Difference  4th Difference  5th Difference 
 
This student written piece of work is one of many that can be found in our GCSE Hidden Faces and Cubes section.
Found what you're looking for?
 Start learning 29% faster today
 150,000+ documents available
 Just £6.99 a month