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

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The discussion centers on various methods to efficiently sum the integers from 1 to 100, ultimately leading to the conclusion that the sum is 5050. Several participants share their approaches, including the classic pairing method where numbers are arranged to highlight their symmetrical relationships, resulting in pairs that each sum to 101. This leads to the formula (n(n + 1))/2, which simplifies to 5050 for n = 100. Alternative visualizations are presented, such as using square arrays and staircase models to illustrate the summation process. Participants also explore more complex summation scenarios, including adding fractions like 1 + 1/n, which relates to harmonic numbers. The conversation touches on the historical anecdote of Gauss, who famously solved the problem as a child, and concludes with a discussion on the lack of a simple closed-form expression for the sum of harmonic series. Overall, the thread emphasizes the elegance and simplicity of arithmetic progressions in calculating sums.
  • #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.
 
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