• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  • Level: GCSE
  • Subject: Maths
  • Word count: 1528

I am going to investigate Pythagorean triples where the shortest side is an odd number and all 3 sides are positive integers - I will then investigate other families of Pythagorean triples to see if Pythagoras’ theorem (a²+b²=c²) works

Extracts from this document...

Introduction

I am going to investigate Pythagorean triples where the shortest side is an odd number and all 3 sides are positive integers. I will then investigate other families of Pythagorean triples to see if Pythagoras' theorem (a�+b�=c�) works. Smallest side Middle side Longest side 3 4 5 +1 (from middle side number) +2 +8 5 12 +4 13 +1 (from middle side number) +2 +8 7 24 +4 25 +1 (from middle side number) +2 +8 9 40 +4 41 +1 (from middle side number) +2 +8 11 60 61 +1 (from middle side number) a�+b�=c� a�+b�=c� a�+b�=c� 3�+4�=5� 7�+24�=25� 11�+60�=61� 9+16=25 49+576+625 121+3600=3721 25=25 625=625 3721=3721 There was only one pattern I noticed in the smallest side, which was the difference of two between each number. However, in the middle side the first difference was +8. This then increased by +4. Therefore, the difference between each number was +4. The longest side patterns were very easy to find. The number was one extra than the number before in the middle side i.e. 4(+1) =5. Smallest side N 1 2 3 4 5 Sequence 3 5 7 9 11 1st differences +2 +2 +2 +2 2n 2 4 ...read more.

Middle

In the middle side, I have noticed that the difference starts with +16 then increases by +8. Therefore, the difference between each number is +8. The longest side does not have a difference. It is achieved by adding two from the number in the middle side. Looking at the patterns, they have been achieved by doubling them from the previous table. Smallest side N 1 2 3 4 5 Sequence 6 10 14 18 22 1st differences +4 +4 +4 +4 4n 4 8 12 16 20 +2 +2 +2 +2 +2 4n+2 Middle side N 1 2 3 4 5 Sequence 8 24 48 80 120 1st differences +16 +24 +32 +40 2nd differences +8 +8 +8 a=8=4 2 4n� 4 16 36 64 100 +4 +8 +12 +16 +20 +4 +4 +4 +4 4n 4 8 12 16 20 +0 +0 +0 +0 +0 4n�+4n Longest Side N 1 2 3 4 5 Sequence 10 26 50 82 122 1st differences +16 +24 +32 +40 2nd differences +8 +8 +8 a=8=4 2 4n� 4 16 36 64 100 +6 +10 +14 +18 +22 +4 +4 +4 +4 4n 4 8 12 16 20 +2 +2 +2 +2 +2 4n�+4n+2 Smallest side (4n+2) ...read more.

Conclusion

Smallest side N 1 2 3 4 5 Sequence 9 15 21 27 33 1st differences +6 +6 +6 +6 6n 6 12 18 24 30 +3 +3 +3 +3 +3 6n+3 Middle side N 1 2 3 4 5 Sequence 12 36 72 120 180 1st differences +24 +36 +48 +60 2nd differences +12 +12 +12 a=12=6 2 6n� 6 24 54 96 150 +6 +12 +18 +24 +30 +6 +6 +6 +6 6n 6 12 18 24 30 +0 +0 +0 +0 +0 6n�+6n Longest Side N 1 2 3 4 5 Sequence 15 39 75 123 183 1st differences +24 +36 +48 +60 2nd differences +12 +12 +12 a=12=6 2 6n� 6 24 54 96 150 +9 +15 +21 +27 +33 +6 +6 +6 +6 6n 6 12 18 24 30 +3 +3 +3 +3 +3 6n�+6n+3 Smallest side (6n+3)� (6n+3) (6n+3) 6n (6n+3) +3 (6n+3) 6nx6n+6nx3+3x6n+3x3 36n�+18n+18n+9 36n�+36n+9 Middle side (6n�+6n) � (6n�+6n) (6n�+6n) 6n� (6n�+6n) +6n (6n�+6n) 6n�x6n�+6n�x6n+6nx6n�+6nx6n 36n4+36n�36n�+36n� 36n4+72n�+36n� Longest side (6n�+6n+3) � (6n�+6n+3) (6n�+6n+3) 6n� (6n�+6n+3) +6n (6n�+6n+3) +3 (6n�+6n+3) 6n�X6n�+6n�x6n+6n�x3+6nx6n�+6nx6n+6nx3+3x6n�+3x6n+3x3 36n4+36n�+18n�+36n�+36n�+18n+18n�+18n+9 36n4+72n�+72n�+36n+9 a�+b�=c� (6n+3)�+ (6n�+6n) �= (6n�+6n+3) � 36n�+36n+9+36n4+72n�+36n�=36n4+72n�+72n�+36n+9 36n4+72n�+72n�+36n+9=36n4+72n�+72n�+36n+9 Conclusion I have predicted correctly. The formulas were as I predicted. I also found out that Pythagoras' theorem did work. Maths Coursework-Mr. Neighbour Jaspreet Athwal 10P ...read more.

The above preview is unformatted text

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Pythagorean Triples essays

  1. 3 Digit Number - Maths Investigations

    =6666 a+b+c+d What if 2 of the 4 digits are the same? If 2 Of The 4 Digits Are The Same : - 1123 1132 1213 1231 1321 1312 2113 2131 2311 3211 3121 +3112 23331 ? 7=3333 Is this the same answer every time 2 of the 4 digits are the same?

  2. Pythagoras Theorem

    C 3 4 5 6 8 10 5 12 13 10 24 26 7 24 25 14 48 50 I then tested this new set of numbers, by putting them trough the original theorem: * 6� + 8� = 100 V100 = 10 * 10� + 24� = 676 V676

  1. Beyond Pythagoras

    are Pythagorean triples. With the help of the site, http://www2.math.vic.edu/~fields/puzzle/triples.html I am able to generate around 70 more Pythagoras theorem. Serial No. Shorter Side a Middle Side b Longest side d 1 3 4 5 2 5 12 13 3 7 24 25 4 9 40 41 5 11 60

  2. Math's Coursework: Pythagoras triples.

    I have found a vital pattern. Using the formula I know for triangular numbers I can find a formula for the Pythagorean triples, in consideration of its order. Formula of the Middle Side Value This is the formula of triangular numbers.

  1. BEYOND PYTHAGORAS

    Here they are: 4) 5) Again, the above are Pythagorean Triples because they satisfy the condition and all their sides have a positive integer. Now, I will draw a table showing the results of the first 5 Pythagorean Triples. T S M L P A 1)

  2. Investigate the area of triangle studies including the Pythagorean Theorem and in particular Pythagorean ...

    According to the process of elimination and trying all sides to investigate all possible paths, I must now try 1 side squared. 122 + 13 = 5 This couldn't possibly work, as 122 is already larger than 5, this also goes for 132.

  1. Beyond Pythagoras.

    1 and 25 and check it on the table above the n table. The number will be 25. 2x25+1=51. This is the same as the number on the table so it must be right. Next I will work out a formula to work out the middle side: First I will see if there are any common differences.

  2. Investigating families of Pythagorean triples.

    I finally came up with: 2n2 + 2n This formula could then be factorised to produce: 2n ( n + 1 ) Once I had found the formula for b, finding the formula for b+1 was easy, I just needed to add one, making the formula for c: 2n2 + 2n + 1 Or 2n (n + 1 )

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work