2n² + 2n
Longest Length
This again gave me the formula 2n², because it was all the same on the second line so I halved it and then squared it. The formula, 2n² gave me the sequence 2, 8, 18, 32, 50. The difference between this formula and the original sequence came to 3, 5, 7, 9, and 11. This sequence had a difference of 2 between each number so you got 2n.
This then gave you the formula 2n² + 2n. I then worked this out and it came to:
4, 12, 24, 40, 60
The difference between this and the original formula is 1 so you then get the final formula of:
2n² + 2n + 1
Perimeter
From finding out the difference of this formula the line, which contained the same number, was the second line. This meant that you had to half this number then square it, so I got the number 4n².
I then worked out this formula and then worked out the difference from this formula and the original sequence.
4n² =
Original =
Difference =
This line then gave me 6n so I then had 4n² + 6n. I worked this out and it gave me:
4 x 1² + 6 x 1 = 10
4 x 2² + 6 x 2 = 28
4 x 3² + 6 x 3 = 54
4 x 4² + 6 x 4 = 88
4 x 5² + 6 x 5 = 120
This then gave me 10, 28, 54, 88, 120. I then worked out the difference between this formula, 4n² + 6n, and the original sequence and it had the difference of 2. This meant that the final formula is:
4n² + 6n + 2
Area
The sequence had a difference of 12 on the 3rd line making the formula 2n³. You have to divide the number by six and then cube it. The formula 2n³ equals:
2, 16, 54, 128, 250
I then worked out the difference between this formula and the original sequence.
Difference =
The second line has the difference of six so you half it and then square it, so you get 3n²
I have now got the formula 2n³ + 3n², which is shown below.
The difference between this and the original formula is shown below.
The difference is 1, which means the formula is + 1n or +n. So the final formula is:
2n³ + 3n² + n
Final Formulas for odd triples
Even Triples
I have now moved on to find out right-angled triangles with an even positive integer for the shortest side. The smallest number, which I have found to be the smallest number, is 6. Below shows a table of all of the even triples I have found.
When I was finding out even triples I could not find any for any number of a multiple of 8, as indicated in the colour turquoise.
I have noticed that some of the middle and longest sides have a difference of 2 and others 4. Also they are in groups of three without the multiples of 8 and the difference goes 2, 4, 2 down the table.
I have placed all the even triples, which have a difference of 2 between the middle and longest length in the table below.
From this table it also shows that the shortest side goes up by 4 each time.
Shortest Length
The difference between the sequence gives you 4n. This formula then gives you:
4, 8, 12, 16, 20
The difference between this formula and the original sequence is 2 and so the final formula is:
4n + 2
Middle Length
The difference between the sequence gives you 4n² as you have to half it then square the number on the second line. This formula then gives you: 4, 16, 36, 64, 100.
The difference of this formula and the original sequence is shown below.
Difference =
This difference gives you 4n so the final formula for the middle length is:
4n² + 4n
Longest Length
The difference between this sequence gives you 4n², as you have to half it then square it because it is on the second line. This formula then gives you: 4, 16, 36, 64, 100.
The difference of this formula and the original sequence is shown below.
Difference =
The formula now is 4n² + 4n, this is shown below.
The difference between this formula and the original sequence is shown below.
This then makes the final formula for the longest length:
4n² + 4n + 2
Perimeter
This gives you the first part of the formula, 8n². You got this by halving the number 16 and then squaring it. The results of this formula are shown below.
I then have found out the difference between the original sequence and formula, 8n², as you can see below.
This difference gives me 12n, so the formula is now 8n² + 12n, this is shown below.
The difference between the original sequence and the formula is now 4 for each stage, so the final formula is:
8n² + 12n + 4
Area
The difference of this sequence is 48 on line 3. This then means that you have to divide by three and then square the number, so you get 8n³. This formula is shown below.
As this is not the same as the original sequence you will need to difference it again. This is shown below.
Difference =
This gives you 12n² because you have to half the end number then square it. This then gives you the formula 8n³ + 12n². The results of this formula are shown below.
The difference of the original sequence and the original formula are shown below.
The difference then gives you 4n, which can then be added to finalise the formula, which is:
8n³ + 12n + 4n
Final Formulas for even triples with a difference of 2
The table below shows even triples with a difference of 4 between the middle and longest lengths.
From this table it shows that the shortest length increases by 8 and the difference between the middle and longest length is 4.
Shortest Length
This difference means that it is 8n. Below I have worked out 8n to see if it is the same as the original sequence.
This is not the same as the original sequence but it does have a difference of 4 for each term, so the final formula is:
8n + 4
Middle Length
This difference means that it is 8n² because as the number is the same on the second line you have to half it then square it. The results of this formula are shown below to see if they match the original sequence.
This does not match the original sequence so I then have to difference the original sequence by the formula, as shown below.
This difference is 8n so it then finalises the formula to make:
8n² + 8n
Longest Length
From finding out the difference it tells you that the beginning of the formula is 8n², as you half the end number then square it. The formula is shown below and then I have found out the difference between the original sequence and the formula.
Difference =
I now have the formula 8n² + 8n. I have shown the results of the formula below.
This is still not the same as the original sequence so below shows the difference between them.
This gives me the final part to the formula, so it is:
8n² + 8n + 4
Perimeter
From finding out the difference it shows that it is 16n², as you have to half 32 then square it.
The formula is shown below and then I have found out the difference between the original sequence and the formula.
16n² =
Difference =
From working out the difference it tells me it is 24n. I have then worked out the formula 16n² + 24n and then found out the difference between the original sequence and the formula.
16n² + 24n =
The difference between the original sequence and the formula is 8, so the final formula is:
16n² + 24n + 8
Area
As the final line is the third line I had to divide it by 6 and then cube it, so it is 32n³. I have shown the results of the formula below and worked out the difference from the original sequence.
32n³ =
Difference =
This difference makes it 48n² as you have to half it then square it. Below I have shown the results of the formula, 32n³ + 48n², and then I have found out if the difference between the original sequence and the formula to see if it matches.
32n³ + 48n² =
Difference =
From this difference it is 16n, so it can then complete the formula so it is:
32n³ + 48n² + 16n
Final Formulas for even triples with a difference of 4
Below I have listed all the final formulas
Final Formulas for odd triples
Final Formulas for even triples with a difference of 2
Final Formulas for even triples with a difference of 4
From studying these tables I have noticed that each formula for the same length is doubled each time. Also the area is multiplied by four each time.
I can also simplify the formulas so they follow the formula : a² + b² = c²
Odd Triples
(2n + 1)² + (2n² + 2n)² = (2n² + 2n + 1)²
However to see if the above formula is a balanced equation I will have to simplify out the brackets.
(2n + 1)² + (2n² + 2n)² = (2n² + 2n + 1)²
(2n + 1) (2n + 1) + (2n² + 2n) (2n² + 2n) = (2n² + 2n + 1) (2n² + 2n + 1)
4n² + 2n + 2n + 1 + 4n + 4n³ + 4n³ + 4n² = 4n + 4n³ + 2n² + 4n³ + 4n² + 2n + 2n² + 2n + 1
As I have simplified the brackets I will now see if they cancel each other out and then I will see what I will end up with.
4n² + 2n + 2n + 1 + 4n + 4n³ + 4n³ + 4n² = 4n + 4n³ + 2n² + 4n³ + 4n² + 2n + 2n² + 2n + 1
4n² = 2n² + 2n²
4n² = 4n²
As you can see from the algebraic equation it balances out so it is equal on each side of the equation. This means that the short side and the middle side equal the longest side, so the equation works.
Even Triples
Difference of 2
(4n + 2)² + (4n² + 4n)² = (4n² + 4n + 2)²
However to see if the above formula is a balanced equation I will have to simplify out the brackets.
(4n + 2)² + (4n² + 4n)² = (4n² + 4n + 2)²
(4n + 2) (4n + 2) + (4n² + 4n) (4n² + 4n) = (4n² + 4n + 2) (4n² + 4n + 2)
16n² + 8n + 8n + 4 + 16n + 16n³ + 16n³ + 16n² = 16n + 16n³ + 8n² + 16n³ + 16n² + 8n + 8n² + 8n + 4
As I have simplified the brackets I will now see if they cancel each other out and then I will see what I will end up with.
16n² + 8n + 8n + 4 + 16n + 16n³ + 16n³ + 16n² = 16n + 16n³ + 8n² + 16n³ + 16n² + 8n + 8n² + 8n + 4
16n² = 8n ² + 8n²
16n² = 16n²
As you can see from the algebraic equation it balances out so it is equal on each side of the equation. This means that the short side and the middle side equal the longest side, so the equation works.
Difference of 4
(8n + 4)² + (8n² + 8n)² = (8n² + 8n + 4)²
However to see if the above formula is a balanced equation I will have to simplify out the brackets.
(8n + 4) (8n + 4) + (8n² + 8n) (8n² + 8n) = (8n² + 8n + 4) (8n² + 8n + 4)
64n² + 32n + 32n + 16 + 64n + 64n³ + 64n³ + 64n² =
64n + 64n³ + 32n² + 64n³ + 64n² + 32n +32n² +32n + 16
As I have simplified the brackets I will now see if they cancel each other out and then I will see what I will end up with.
As I have simplified the brackets I will now see if they cancel each other out and then I will see what I will end up with.
64n² = 32n² + 32n²
64n² = 64n²
As you can see from the algebraic equation it balances out so it is equal on each side of the equation. This means that the short side and the middle side equal the longest side, so the equation works.