Beyond Pythagoras

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Sam Caddy

Beyond Pythagoras

In this piece of coursework I will be exploring Pythagorean triples which are beyond Pythagoras.

Pythagoras is where you have a right angled triangle and you know the values of sides a + b but you don’t know what side c is (hypotenuse). To calculate the hypotenuse you can square sides a + b and add the two answers together as one total. This total is equal to c2 so all you have to do now is find the square root of the total and you have worked out side c.

A Pythagorean triple is where the first two sides of a triangle (a + b) suit this equation: a2 + b2 = c2. For example lets say a=1, b=2, c=3. Now let’s try this in a Pythagoras equation: a2 + b2 = c2 (12 + 22 = 32) this is not correct!!! 12 + 22 = 5 we should know that 32 = 9!!!! From this we should acknowledge that the 3 numbers I used don’t fit the equation a2 + b2 = c2, this therefore means they are not Pythagorean triples!!!!

So from this you should notice that a2 + b2 = c2, if a=3, b=4, and c=5 (32+44 = 52) we can see that 32 + 42= 52(25) this therefore means these numbers make a Pythagorean triple.

I have been given 3 sets of Pythagorean triples from family 1 to analyse:

        

I’ve been told that family 1 of the Pythagorean triples has the following features:

●smallest side is odd

●the longest side is one more than the middle side

●on the middle side you add 4 more on than the last time

If I wanted to work out more Pythagorean triples for family 1 I can search for patterns from the data I have been provided with.

The easiest way to work out if there is a pattern is to find a formula which will enable you to work out any part of the sequence!

As you can see in the table below there are 5 sets of Pythagorean triples for family 1. I was given 3 Pythagorean triples to begin with from family 1 but I have worked out and added two additional Pythagorean triples to the table (which are in green)

Family 1

                                                                                                                                                       

As you can see I have spotted patterns in family 1 I should be able to find a formula if I separate each side of family 1. I will make tables for the sequences of each side of any family, or patterns when I’m ever trying to find the nth term of anything.

 

Small

        

From looking at this table I have noticed that there is a pattern between how much you how much you add to get to the next sequence, in this case it’s always +2. Also when working out this formula I noticed that the difference between each sequence which in this sequence is repeatedly +2 goes before n in the final formula. So I have 2n also notice that by working out the imaginary 0th term I can add the sequence number from that term and add this to the 2n, it looks like I’ve just successfully worked out a formula!!! This method works when working out the nth term for something which always uses the same figure to get to the next sequence.

I worked out that the formula = 2n + 1

I will make a prediction to prove my formula works:

I predict that for no. 10 the sequence will be (10x2) + 1 = 21.

How can I prove I’m right?

Looking for patterns in the sequence helps, for instance in this table for the small side of family 1 I noticed that you add two on every time you move up the sequence. I’ll test this against what my formula predicts:

Join now!

From no.5 the sequence = 11, so for no.6 the sequence will = 13, for no.7 the sequence will = 15, for no.8 the sequence will = 17, for no.9 the sequence will = 19, finally for no. the sequence will = 21

I was correct!!!! The formula does work.

Now I’ve worked out a successful formula for the small side of family 1 I will follow the same methods I used to work out formulas for the middle & longest sides of family 1.

Middle

From looking at this ...

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