/sci/ - Science & Math » Thread #12998977

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Anonymous Mon 19 Apr 07:02:49 2021 No.12998977 View Reply Original Report

Quoted By: >>12999059 >>12999084 >>12999258 >>12999282 >>12999336

How come (x^2-x)/2 is always equal to the sum of all integers from 0 to x-1?
Pic unrelated

Anonymous

Anonymous Mon 19 Apr 2021 07:03:56 No.12998982 Report

Quoted By: >>12999051

Like ive seen the proof but i just dont get intuitively why this works

Anonymous

Anonymous Mon 19 Apr 2021 07:28:25 No.12999051 Report

Quoted By:

>>12998982
Ok i lied i dunno the proof plz help i dont wanna fail

Anonymous

Anonymous Mon 19 Apr 2021 07:30:33 No.12999059 Report

Quoted By:

>>12998977
Anon, this proof is left to be solved by the reader.

Anonymous

Anonymous Mon 19 Apr 2021 07:39:24 No.12999084 Report

Quoted By:

>>12998977
Not doing your homework faggot

Anonymous

Anonymous Mon 19 Apr 2021 08:53:47 No.12999258 Report

Quoted By:

>>12998977
Look at the differences between the squares (1, 3, 5, 7, ...). It will probably help with your understanding of whatever the fuck you have to do here.

Anonymous

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Anonymous Mon 19 Apr 2021 09:00:21 No.12999282 Report

Quoted By: >>12999289

>>12998977
1+2+3+... stacked can form a pyramid with base width x-1, which has surface equal to half of the rectangle with height x.

The same visualization also shows that this number also equals (x choose 2).

Anonymous

Anonymous Mon 19 Apr 2021 09:02:21 No.12999288 Report

Quoted By:

Note that
$<br />\sum_{k=0}^{x-1} k = \sum_{k=0}^{x-1} (x-k-1)<br />$
You just reverse the order of summation. Now add the LHS to both sides to get
$2 \sum_{k=0}^{x-1} k = \sum_{k=0}^{x-1} (x-k-1) + \sum_{k=0}^{x-1} k = \sum_{k=0}^{x-1} (x-1) = x (x-1) = x^2 - x<br />$
Now divide by 2 and you're done.

Other possible proofs include induction or differentiating the formula for a finite geometric sum.

Anonymous

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Anonymous Mon 19 Apr 2021 09:02:23 No.12999289 Report

Quoted By:

>>12999282
here the same with the rectangle drawn explicitly

Anonymous

Anonymous Mon 19 Apr 2021 09:22:15 No.12999336 Report

Quoted By:

>>12998977
take 10 for example
1+2+3+4+5+6+7+8+9+10
1+10 = 11
2+9 = 11
3+8 = 11
4+7 = 11
5+6 = 11
11*5 = 55
or x/2 * (x+1)

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