Beyond Pythagoras

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Mahmoud Elsherif               Beyond Pythagoras                  P.1

Pythagoras Theorem is a2+b2= c2 ‘a’ is being the shortest side, ‘b’ being the middle side and ‘c’ being the longest side (hypotenuse) of a right angled triangle. 

The numbers 3,4,5 satisfy this condition and so

32+ 42=52

Because 32= 3*3=9

42=4*4=16

52=5*5=25

32+ 42=52

9+16=25

25=25

This proves Pythagoras Theorem goes with the right angled triangle with the numbers 3,4,5. Next I shall prove that Pythagoras’s Theorem applies to 5,12,13 right angled triangle.

52+122=132

Because 52= 5*5=25

122= 12*12=144

132= 13*13=169

Mahmoud Elsherif                        Beyond Pythagoras                         P.2

This satisfies the Theorem of Pythagoras’s goes with these numbers 5,12,13. Finally I shall prove that Pythagoras’s Theorem applies to 7,24,25 right angled triangle.

72+ 242=252

Because 72= 7*7=49

242= 24*24= 576

252=25*25=625.

So

a2+b2=c2

72+242=252

49+576=625

This proves Pythagoras Theorem goes with the right angle triangle with the sides 7,24,25

Mahmoud Elsherif                         Beyond Pythagoras                        P.3

I shall find the prediction of the shortest side first.

3,5,7

It goes up in 2 so in my conclusion so it will become

3,5,7,9,11,13

Now I will find the difference between them.

The difference is 2

Next I shall find the prediction of the middle side next.

4,12,24

It goes up by 4,8,12. So in my conclusion I think it will become 4,8,12,16,20,24

So it will be 4,12,24, 40, 60 84.

The difference is 4,8,12. Now I shall find the difference and it is n*4

Next I shall find the prediction of the longest side next.

5,13,25

It goes up by 4,8,12. So in my conclusion I think it will become 4,8,12,16,20, 24

So it will be 5,13,25,41,61 85.

The difference is 4,8,12. Now I shall find the difference and it is n*4.

First I shall find the nth term of the shortest side.

So 2/1=2, so it’s 2n. Substituting in we get for 2n

If n=1     2 and the first term is 3 so a difference is +1

If n=2            4 and the first term is 5 so a difference is +1

If n=3     6 and the first term is 7 so a difference is +1

Therefore the rule for finding the nth term of the shortest side is:

2n+1

Mahmoud Elsherif                Beyond Pythagoras                                P.4

Next I shall find the nth term of the middle side.

4/2=2

So it’s 2n2.

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Therefore the rule for finding the n
th term of the middle side is:

2n2+2n

Next I shall find the nth term of the longest side.

4/2=2

So it’s 2n2.

So we have so far 2n2+2n. Substituting in we get for 2n2+2

If n=1          4 and the first term is 5 so a difference of +1

If n=2    12 and the first term is 13 so a difference of +1

If n=3    24 and the first term is 25 so a difference of +1

Mahmoud Elsherif                         Beyond Pythagoras                         P.5

So the nth term is:

2n2+2n+1

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