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Beyond Pythagoras

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Introduction

Beyond Pythagoras Sides (smallest number) 2 + (middle number) 2 =(largest number) 2 To test this: Perimeter Smallest number + middle number + largest number = perimeter To test this: Area 1/2 x smallest number x middle number = area To test this: Length of the shortest side Length of the middle side Length of the longest side Perimeter Area 3 4 5 12 6 5 12 13 30 30 7 24 25 56 84 9 40 41 90 180 11 60 61 132 330 13 84 85 182 546 15 112 113 240 840 17 144 145 306 1224 I managed to fill in the rest of the sides without using a formula by firstly for the shortest side add on two units to the value before. For the middle side the difference between each value is positive four units. For the longest side I added I=one unit to the value of the middle side in the same sequence. I have also noticed that the middle values are the triangle numbers multiplied by four. Also the second difference between each of the perimeter values is eight units. In the area the third difference between each of the values is twelve units. Shortest side I have worked out the nth term formula for the shortest side. ...read more.

Middle

From my class work I know I can work out the nth term formula using this equation: "a+(n-1)d+1/2(n-1)(n-2)c". * "a" is the value of the first term in the sequence. * "d" is the value of the first difference. * "c" is the value of the change in difference between one difference and the next. Applying this formula you get: "6+(n-1)24+1/2(n-1(n-2)12" simplified you get "6n2+6n-6" To test this formula I will apply the formula to the 1st and 2nd terms. 1st term: 2nd term: 6x12+6x1-6=6 6x22+6x2-6=30 This proves that my formula is correct. Now I will generalize everything. I have investigated some more Pythagoras triples and found out that the Pythagorean triple 9,12,15 does not satisfy my favourite triples that are even. I have looked on the internet and found a list of Pythagorean triples and found that 9,12,15 is a multiple of the famous 3,4,5 Pythagorean triple. Multiplying this Pythagorean triple will also give you other Pythagorean triples. X2 =6,8,10 X3 =9,12,15 X4 =12,16,20 X5 =15,20,25 I am now going to investigate even pythagoras triples. A Pythagorean Triple is a triple of natural numbers (a,b,c) such that: "c2 = a2 + b2. It can be shown (this is a good exercise!) that all pythagorean triples can be written as: "a = x2 - y2, b = 2xy, c = x2 + y2" where x > y > 0 are natural numbers. ...read more.

Conclusion

This is how I did it: This sequence has a second difference, eight between each term. From my class work I know I can work out the nth term formula using this equation: "a+(n-1)d+1/2(n-1)(n-2)c". * "a" is the value of the first term in the sequence. * "d" is the value of the first difference. * "c" is the value of the change in difference between one difference and the next. Applying this formula you get: "24+(n+1)16+1/2(n-2)(n-2)4" simplified you get "2n2+10n+12" To test this formula I will apply the formula to the 1st and 2nd terms. 1st term: 2nd term: 2x12+10x1+12=24 2x2+ 10x2+12=40 This proves that my formula is correct. Area I have worked out the nth term formula for the middle side. This is how I did it: This sequence has a third difference, twelve between each term. From my class work I know I can work out the nth term formula using this equation: "a+(n-1)d+1/2(n-1)(n-2)c". * "a" is the value of the first term in the sequence. * "d" is the value of the first difference. * "c" is the value of the change in difference between one difference and the next. Applying this formula you get: "624+(n-1)36+1/2(n-1(n-2)6" simplified you get "3n +27n-6" To test this formula I will apply the formula to the 1st and 2nd terms. 1st term: 2nd term: 3x1 +27x1-6=6 3x2 +27x2-6=30 This proves that my formula is correct. ...read more.

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