# Beyond Pythagoras

BEYOND PYTHAGORAS

By: Megan Garibian

10A

What this coursework has asked me to do is to investigate and find a generalisation, for a family of Pythagorean triples. This will include odd numbers and even numbers.

I am going to investigate a family of right-angled triangles for which all the lengths are positive integers and the shortest is an odd number.

I am going to check that the Pythagorean triples (5,12,13) and (7,24,25) cases work; and then spot a connection between the middle and longest sides.

The first case of a Pythagorean triple I will look at is:

The numbers 5, 12 and 13 satisfy the connection.

5² + 12² = 13²

25 + 144 = 169

169 = 13

The second case of a Pythagorean triple I will look at is:

The numbers 7, 24 and 25 satisfy the connection.

7² + 24² = 25²

49 + 576 = 625

625 = 25

There is a connection between the middle and longest side. This is that there is a one number difference.

So if M= middle and L= longest

L = M + 1

I am going to use the triples, (3,4,5), (5,12,13) and (7,24,25) to find other triples. Then I will put my results in a table and look for a pattern that will occur. I will then try and predict the next results in the table and prove it.

There is a clear pattern between the middle and longest side.

There is also a sequence forming.

n = 1                S = 3                M = 4                L = 5

n = 2                S = 5                M = 12                L = 13

n = 3                s = 7                M = 24                L = 25

2n +1                            Sxn +n                        M+1

I have established a connection between n and S, S and M, and, M and L.

I can see if I multiply n by 2 and add 1 to it I get S.

I can see if I multiply S by n and add n to it I get M.

I can see if I add 1 to M I get L.

In another way S, M, and L are forming a sequence. I know how to find the n   term for a sequence so I applied this knowledge and came up with a formula.

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