Check if correct formula:
(Check On Page 11)
Finding the nth term for the 3rd difference:
In order to find out the nth term for the 3rd differences, the requirement is to subtract the 1st fraction from the 2nd fraction (the smaller fraction from the bigger fraction) of the 2nd differences.
-
=
=
= (The general formula for 3rd difference)
Check if correct formula:
(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:
(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:
(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:
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 formula y = an3 + bn2 + cn + d
* Note: y in the above formulas is the denominator. *
Solving the constants b and c
In order to solve the constants b and c, we have to use the table above in order to solve the equations simultaneously.
Formula No 2 – 1
= 4b + 2c + d = 26.0
b + c + d = 11.5
___________________
3b + c = 14.5 (Call this I.)
___________________
Formula No 3 – 2
= 9b + 3c + d = 46.5
4b + 2c + d = 26.0
___________________
5b + c = 20 .5 (Call this II.)
___________________
Through the above simultaneous equations, d has been eliminated.
Formula No II. – I.
= 5b + c = 20.5
3b + c = 14.5
________________
2b = 6
b = 3
________________
-
In order to find c, we have to substitute b = 3 in to the equation II.
15 + c = 20.5
c = 20.5 – 15.0
c = 5.5
-
In order to find d, we have to substitute b = 3 and c = 5.5 into Formula No 3
9b + 3c + d = 46.5
27 + 16.5 + d = 46.5
d = 46.5 – 27.0 – 16.5
d = 3
The denominator formula:
Incorporating [a = , b = 3, c =5.5, d = 3] into the formula y = an3 + bn2 + cn + d
= y = n3 + 3n2 + 5.5n + 3
* 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.
The general term for the fractions is:
Check if correct formula:
(Check on page 11)
Checking Sheet