klimenkov
  • klimenkov
Prove for a medal. The sum of the first \(n\) terms of the arithmetic sequence \(1+2+3+\cdots+n\). Try to find as more proofs as you can.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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experimentX
  • experimentX
|dw:1349108754795:dw|
anonymous
  • anonymous
What statement is to be proved? That a sum exists?
experimentX
  • experimentX
|dw:1349108832681:dw|

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experimentX
  • experimentX
|dw:1349108956157:dw|
anonymous
  • anonymous
The method I learned was that n + 1 = n-1 + 2 = n-2 + 3 = . . . There are n/2 such pairs of equal sums, so the total sum is (n/2)(n+1)
anonymous
  • anonymous
Simplest Proof other than those given: \[\sum_{r=1}^{n}r = A\] \[A = \sum_{r=1}^{n}(n+1-r)\] \[Adding, 2A =(n+1)\sum_{1}^{n}1 = (n+1)(1+1+1+1+1.. ..(\times n))\] \[=> A = n(n+1)/2\]
klimenkov
  • klimenkov
Any other ideas?
anonymous
  • anonymous
You are writing a book..?:-)
klimenkov
  • klimenkov
No. It is interesting to know.
anonymous
  • anonymous
Are other methods actually needed? The one I use is straight-forward, easy to understand, and I can teach it to children in about a minute.
anonymous
  • anonymous
I understand the pull of curiosity, though.