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 \(S = \frac{1}{6} n(n+1)(2n+1)\) #proofinatoot (via Jeremy Kun)

But for ellipses

We'd have no eclipses

#groot

#reckonthatmustbetrue

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! http://chalkdustmagazine.com/write-for-us/

@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. #getofmylawn

- Website
- colinbeveridge.co.uk

- Location
- Dorset, UK

- Latest book
- https://amzn.to/2MUuEbs

- Pronouns
- He/his

A mathematician with nothing to prove.

Joined May 2017