Proof 

Let \(S = \sum_{r=1}^n r^2\).

Then \(S = \\
1+\\
2+2+\\
3+3+3+\\
\cdots \\
n+n+n+\cdots+n\)

Also, \(S = \\
n+\\
n+(n-1)+\\
n+(n-1)+(n-2)+\\
\cdots \\
n+(n-1)+(n-2)+\cdots+2+1\)

And \(S = \\
n+\\
(n-1)+n+\\
(n-2)+(n-1)+n+\\
\cdots \\
1+2+3+\cdots+(n-1)+n\)

So \(S+S+S = \\
(2n+1)+\\
(2n+1)+(2n+1)+\\
(2n+1)+(2n+1)+(2n+1)+\\
\cdots \\
(2n+1)+(2n+1)+(2n+1)+\cdots+(2n+1)+(2n+1)\)

i.e., \(3S = (2n+1) \times \frac{n(n+1)}{2}\) or \(S = \frac{1}{6} n(n+1)(2n+1)\) (via Jeremy Kun)

“Why isn’t the new year on winter solstice?”

The answer, honestly, is that the Romans had no fucking idea how to run a calendar.

Having trouble with MastodonToTwitter.py - don't know where to turn for help.

File
"/usr/local/lib/python3.4/dist-packages/mastodon/Mastodon.py", line 963, in __api_request

if url['rel'] == 'next':

KeyError: 'rel'

Help?

A young man defeats several martial arts masters by inventing a martial art that depends solely on his infallibly good luck. He names it Bullshitsu.
#writingprompts #writing

If any tooters want to get an article in issue 06 of Chalkdust, get writing! chalkdustmagazine.com/write-fo

Is \( \sum_{r=100}^{0} r \):

a) 5050
b) 0
c) -5050
d) other

I have an Opinion, but I don't want to share it until I know whether it's controversial (or worse yet, Wrong.)

@ColinTheMathmo I would consider a delay in email a feature, not a bug. People seem to confuse email with instant messaging. Email is supposed to be asynchronous. I guess some instant messaging tools are also supposed to be asynch, but apparently everything should be like SMS.

Is there an easy way to turn off the 'bloop' notifications? I can't see it in the settings.

Let \(C_1\) and \(C_2\) be circles that intersect at right angles, having \(P_1\) and \(P_2\) as their centres. Then the circle with diameter \( \vec{P_1 P_2} \) has a quarter of the area of the sum of the areas of \( C_1 \) and \( C_2 \).

Mathstodon

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