The formula to find the first n positive n integers is actually very simple.
Sn= n^2
1.To find the formula I first found the formula for the arithmetic sequence. The final formula was Tn= 2n-1, so with this formula I could actually get the last number of the sequence.
2.Then I related the Tn formula with Gauss´s formula.
In the formula Sn= n/2(T1 +Tn) I replaced the value for Tn so Sn= n/2(T1 + (2n-1)).
3.Since T1 = 1 then the formula would be Sn= n/2(1 +2n-1)
4.Cancel out the ones and the formula ends as Sn= n/2(2n) or Sn= (2n^2)/2, cancel the two and I end up with Sn=n^2.
5.Example: 1 +3+5+7+9 is the series until the fifth term. S5= 5^2=25. Eventually 1+3+5+7+9=25
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2 comments:
Nicely done, sir.
Your approach is elegant.
Just for fun, try proving your formula using the difference analysis technique.
Wow Jaime. Nice. It is interesting to see how simple it is.
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