How can you easily add the numbers from 1 to 100?

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SUMMARY

The discussion centers on the mathematical approach to summing the integers from 1 to 100, with the definitive formula being S = n(n + 1)/2, where n is the last integer in the sequence. Participants highlight various methods to visualize and derive this sum, including pairing numbers and using geometric representations. The final result of the sum is established as 5050, confirmed through multiple perspectives, including arithmetic progression and triangular number concepts. Additionally, the conversation touches on extending these principles to non-integer sequences and harmonic numbers.

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  • Understanding of arithmetic progression
  • Familiarity with basic algebraic manipulation
  • Knowledge of geometric representations in mathematics
  • Concept of harmonic numbers and their approximations
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  • Explore the derivation of the formula for the sum of the first n integers
  • Learn about harmonic series and their applications
  • Investigate geometric interpretations of mathematical sequences
  • Study advanced summation techniques and their proofs
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Mathematicians, educators, students learning arithmetic series, and anyone interested in mathematical problem-solving techniques.

  • #31
Air said:
1+100=101, 2+99=101, 3+98=101 etc. There are 50 pairs of these so it is 101 times 50 which equals 5050.

Finally someone who didn't make the problem more complicated than was necessary and explained it the same exact way that Euler did it (or was it Gauss?).
 
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  • #32
i have somee good, yet simple brain teaasers, if any of you could give me advice on how to post them, that's be great, but for now, i'll just write a cool one here:a homeless man can collect 5 cigar butts to make 1 cigar. he finds 25 cigar butts, how many cigars can he make(hint:he smokes them all as soon aas he makes them>(my o don't work>





make some kinda reply and ill getcha the answer
 
  • #33
Six?

To post a new teaser, go up one level to the https://www.physicsforums.com/forumdisplay.php?f=33 and click the "New Topic" button near the top left corner of the page.
 
  • #34
The asymptotic behavior has an interesting geometric interpretation:
http://img201.imageshack.us/img201/5534/staircasesumsd2.png

Now here is a related brainteaser: what is the sum
1^2 + 2^2 + 3^2 + 4^2 + ... + n^2
in the limit as n is large?
 
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  • #35
Edit: I probably should have posted this with a spoiler alert:

\fbox{\color{white}n^3/3}... (LaTeX with white text... click to view source.)
Fill a portion of a cube with a stack of sequential squares... the analogue of your diagram. It generalizes to higher dimensions.
 
Last edited:

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