1) Sum of i as i goes from 1 to n : 1 + 2 + 3 + ... + n
2) Add the summation to itself, but think of adding the second summation backwards:
1 + ... + n
n + ... + 1
3) If you think of the first Sum as (i), then the second sum becomes (n - i + 1)
4) So 2 * the sum is (i) + (n - i + 1) or (n + 1) for each item in the series
5) Since there are n items in the series, the 2 * sum value is n * (n + 1)
6) for a single series, we end up with (n * (n + 1) / 2)
It's pretty straightforwards to convert this proof for a series of arbitrary interval.
11 comments:
between 1777-1855, carl friedrich gauss discovered your theorem, when he was in junior scholl, so you didnt discover anything you dumbass!
how can you call him a dumbass when 1) you spelt school wrong
2) i found this site helpfull as i couldnt find the equation anywhere else
and 3) you didnt post the formulae yourself now did you
thx a lot, helpful, disregard first comment...
Cheers! The one proof I needed but didn't get taught, just in time for my exam tomorrow! Also, ignore whoever left the first comment since they clearly can't spell or read your second sentence. Besides, I don't reckon this Carl Gauss was up so great if he spent 78 years in junior "scholl" :)
You did a great job here! A cleaner approach would be a proof by induction. That is, you start by assuming the case for some integer k is true. If you can prove that your relationship holds for the (k+1) case, you are home free. What you did is basically the groundwork for a proof. It is clear that you have the right idea, you just need to lock down the logic a bit more. Otherwise, you can poke holes in the logic. Yours is the proof that this works for some integer n, but does it work for any n?
Also, it is true that Gauss discovered this when he was a child; but if Gauss is the standard, we are all "dumbasses".
thanks! :)
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