Beyond Pythagoras

I have established a connection between n and S, S and M, and, M and L.

I can see if I multiply n by 2 and add 1 to it I get S.

I can see if I multiply S by n and add n to it I get M.

I can see if I add 1 to M I get L.

In another way S, M, and L are forming a sequence. I know how to find the n   term for a sequence so I applied this knowledge and came up with a formula.

By: Megan Garibian

Prediction:

 I am going to use the formula I found to predict the next set of results.

S=  2n +1                M=  Sxn +n                L=  M+1

      2x4 +1                       9x4 +4                     40+1

      8 +1                       36 +4                L = 41                    

S = 9                      M = 40

This means my prediction is, that the next triple will be:

Shortest side = 9

Middle    side = 40

Longest  side = 41

I will prove my prediction by using Pythagoras thermo.

9²+40²=41²

81 + 1600 = 1681           1681 = 41

My prediction was correct. That means my formula works.

Now I am going to try my formula a few more times, and also to check that the new data in the table is correct.

Then I will use the Pythagoras thermo to see if the triples will work.

 

By: Megan Garibian

I am going to test out the triples (13,84,85), (17,144,145) and (21,220,221)

The first triple I will test out will be: (13,84,85) with the n   term being 6.

S=  2n +1                M=  Sxn +n                L=  M+1

      2x6 +1                       13x6 +6                    84 +1

       12 +1                       78 +6                L = 85            

S = 13                     M = 84

13²+84²=85²

169 + 7056 = 7225                7225 = 85

The second triple I will test out will be: (17,144,145) with the n   term being 8.

S=  2n +1                M=  Sxn +n                L=  M+1

      2x8 +1                       17x8 +8                     144 +1

      16 +1                       136 +8                L = 145            

S = 17                    M = 144

17²+144²=145²

289 + 20736 = 21025                21025 = 221

The third triple I will test out will be: (21,220,221) with the n   term being 10.

S=  2n +1                M=  Sxn +n                L=  M+1

      2x10 +1                       21x10 +10            220 +1

       20 +1                        210 +10                L = 221            

S = 21                        M = 220

21²+220²=221²

441 + 48400 = 48841                48841 = 221

This proves my formula to be correct again, and it proves that the data in the table is correct.

By: Megan Garibian

Now I am going to find a general rule for the shortest, middle and longest side in algebra.

I am going to show that the algebra is correct by finding other triples using the general rule.

To find a general rule, I am going to use the n   term as 1.

First of all I am going to find the general rule for the shortest side.

S= 2n +1

This is the general rule for the shortest side.

Now I am going to find the general rule for the middle side.

M= Sxn +n

M= (2n +1) xn +n        =         2n²+n+n

                        =         2n²+2n        

So the general rule for the middle side is: 2n²+2n

Finally I am going to find the general rule for the longest side.

L= M+1

L= (2n²+2n) +1        =        2n²+2n +1

So the general rule for the longest side is: 2n²+2n +1

Now I am going to use the general rules that I have found to prove it is correct.

If n = 9

S = 2n +1                M = 2n²+2n                        L = 2n²+2n +1

   = 2x9 +1                    = 2x9² + (2x9)                   = 2x9² + (2x9) +1

   = 18 +1                    = 162 + 18                            = 162 + 18 +1

   = 19                     = 180                           = 181

To see if the triple will work I am going to use Pythagoras thermo.

By: Megan Garibian

19²+180²=181²

361 + 32400 = 32761                32761 = 181

I now am going to find more triples with the shortest side being an odd number, and all three sides are positive integers.

If n = 50

S = 2n +1                M = 2n²+2n                        L = 2n²+2n +1

   = 2x50 +1                    = 2x50² + (2x50)            = 2x50² + (2x50) +1

   = 100 +1                    = 5000 +         52                    = 5000 + 52 +1

   = 101                    = 5052                           = 5053

To see if the triple will work I am going to use Pythagoras thermo.

101²+5052²=5053²

10201 + 25522704 = 25532809                25532809 = 5053

If n = 19

S = 2n +1                M = 2n²+2n                        L = 2n²+2n +1

   = 2x19 +1                    = 2x19² + (2x19)            = 2x19² + (2x19) +1

   = 38 +1                    = 722 + 21                            = 722 + 21 +1

   = 39                               = 743                           = 744

To see f the triple will work I am going to use Pythagoras thermo.

39²+743²=744²

1521 + 552049 = 553536                553536 = 744

By: Megan Garibian

I have finished investigating this family of Pythagorean triples where the shortest side is an odd number and all three sides are positive integers.

I have checked through cases of Pythagorean triples to see if they satisfy the conditions and spotted a connection between the middle and longest side.

Then I used the first three triples in the sequence to find a pattern and to predict the next results.

I also extended the Pythagorean triples to a sequence of 10, found connections between the short, middle and longest sides. I expressed these connections in algebra and gave a reason for it.

I then found a general rule for the shortest, middle and longest sides in algebra, and proved it was correct by using Pythagoras thermo.

Finally I found more triples with the shortest side being an odd number and all the three sides are positive integers.

By: Megan Garibian