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# Pythagorean triplets

Extracts from this document...

Introduction

Thomas Brown        G.C.S.E Maths Coursework        Mr Tims

11R2        Beyond Pythagoras        Pg

Introduction

a2 + b2= c2

## A simple example of this is these numbers: 3 , 4 , 5

Because 32  =  3 * 3 = 9

42 = 4 * 4 = 16

52= 5 * 5 = 25

32 + 42 = 9 + 16 = 25 = 52

This is the 1st Pythagorean Triple

Another example is: 5 , 12 , 13

Another Example is: 7 , 24 , 25

Middle

Length of

Shortest side

Length of middle side

Length of longest side

Perimeter

Area

3

4

5

12

6

5

12

13

30

30

7

24

25

56

84

We can now put these results into a table:

To  find the lengths of the sides of the 4th and 5th triplets I have to try and recognize a pattern        :

### Shortest Side

3

`                      + 2

5

+ 2

7

The difference between the numbers of the shortest side seems to be 2. I can now confidently say that the shortest side in the 4th triple will calculate to be 9.

### Middle side

4

+   8

12+4

+ 12

24

I have found that the difference (2) of the difference (1) is 4,so,

I can estimate that the length of the middle side for triplet 4 will be 40.

24 + 16 = 40

#### Longest Side

Here are the lengths of the longest sides:

Middle                Longest

4                +1             5

12                +1            13

24                +1              25

## smallest side     middle side        longest side          units          sq  units

9                40                        41                90                180

I can now see if these numbers satisfy the condition  a2 + b2= c2

###### This is the 4th Pythagorean triple

I can now put this new result into my table and work out the 5th triple

 Length of Shortest side Length of middle side Length of longest side Perimeter Area 3 4 5 12 6 5 12 13 30 30 7 24 25 56 84 9 40 41 90 180 11 60 61 132 330

Conclusion

Longest side = +1 to middle side

Finding a nth term for the sequences:

Smallest Side

n                         1                    2                  3                4                 5

Smallest side     3                    5                 7                 9                11

Difference2                   2                  2                  2

After studying the grid I have found that the formulae is 2n + 1

Examples :  1 (n) * 2 + 1 = 3

3 (n) * 2 + 1 = 7

5 (n) * 2 + 1 = 11

Middle Side

n                       1                    2                       3                4                5

Middle Side               4                    12             24               40             60

Diff 18                   12                6              20

Diff 2                                        4                4                 4

½ 2nd Diff                                        2                2

This sequence is quadratic therefore we know it will include and n2. Now I know the sequences ½ 2nd difference is 2 I Know the first part to my formulae will be 2n2.

I will now attempt to work out the 2nd part to my formulae by using a table

 Triple 2n2 Middle side Difference between 2n2 and Middle 1 2 4 2 2 8 12 4 3 18 24 6 4 32 40 8 5 50 60 10

I can see now that the difference between n2 and middle side is 2 so I can now say that the formulae will be 2 (n2 ) + 2.

Examples -  2 (1(n)2) + 2n = 4

-  2 (3(n)2) + 2n = 24

-  2 (5(n)2) + 2n = 60

This student written piece of work is one of many that can be found in our GCSE Beyond Pythagoras section.

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