Beyond pythagoras - First Number is odd.

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MATHS COURSEWORK-PYTHAGORAS TRIPLE’S                   09/05/2007

BEYOND PYTHAGORAS

First Number is odd

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

1) The Pythagoras Theorem a2 + b2 = c2, is basically (Smallest Number) 2 + (Middle Number) 2=(Largest Number) 2

a) The numbers 5, 12 & 13 satisfy the condition.

            52 + 122 = 132

Because    52  = 5 X 5= 25

          122  = 12 X 12= 144       

           132  = 13 X 13= 169

and so

          52 + 122 = 25 + 144= 169= 132

        

b) The numbers 7, 24 & 25 satisfy the condition.

            72 + 242 = 252

Because    72  = 7 X 7= 49

          242  = 24 X 24= 576       

           252  = 25 X 25= 625

and so

         72 + 242 = 49 + 576= 625= 252

2) The numbers 3, 4 and 5 can be lengths – in appropriate units – of the sides of a right- angles triangle.

3,4,5 Triangle

Perimeter= 3 + 4 + 5= 12 units

        Area= 3 x 4 x ½= 6 square units

5,12,13 Triangle

Perimeter= 5 + 12 + 13= 30 units

        Area= 5 x 12 x ½= 30 square units

7,24,25 Triangle

Perimeter= 7 + 24 + 25= 56 units

        Area= 7 x 24 x ½= 84 square units

From the first three sequences I have noticed:

  • ‘a’ increases by +2 each term
  • ‘a’ is equal to the term number times 2 then add 1
  • The last digit of ‘b’ is in a pattern 4, 2, 4
  • The last digit of ‘c’ is in a pattern 5, 3, 5
  • The square root of (‘b’ + ‘c’) = ‘a’
  • ‘c’ is always +1 to ‘b’
  • ‘b’ increases by +4 each term
  • (‘a’ x ‘n’) + n = ‘b’


So, the equation I have so far is: 2n2+2n

I will now check if the formula works:


2 x 1² + 2 x 1 = 4
2 x 1 + 2 = 4
2 + 2 = 4
4 = 4

My formula works for the first term, so, I will now check it on the 2nd term.
2 x 2² + 2 x 2 = 12
2 x 4 + 4 = 12
8 + 4 = 12
12 = 12

My formula also works for the 2nd term. If it works for the 3rd term I can safely say that 2n2 + 2n is the correct formula.

2 x 3² + 2 x 3 = 24
2 x 9 + 6 = 24
18 + 6 = 24
24 = 24

My formula also works for the 3rd term. I am now certain that 2n2 + 2n is the correct formula for finding the middle side.

Join now!

Middle Side = 2n² + 2n

Longest Side:

I know that there is only a difference of 1 between the middle side and the longest side. So:
(Middle side) + 1 = Longest side.
2n² + 2n + 1 = Longest Side
I am certain that this is the correct formula. Just in case, I will check it using the first 3 terms.
2n² + 2n +1 = 5
2 x 1² + 2 x 1 + 1 = 5
2 + 2 + 1 = 5
5 = 5

The formula works for the first term.
2n² + 2n +1 = 13
2 x 2² + 2 ...

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