Question

Whom of you can explain it to me?

http://jeremykun.files.wordpress.com/2011/06/triangle-proof.png

Answer 1

http://mathoverflow.net/questions/8846/proofs-without-words/69756#69756

Answer 2

If it's fairly easy and you think I am not pushing myself please do tell me, instead of giving out the answer.

Answer 3

BTW: Nice site.

Answer 4

@Ishaan94 - do you know how to get the sum of all the components in the last triangle?

Answer 5

No ._.

Answer 6

wait

Answer 7

Is it like this?

(2n+1)+2(2n+1)+3(2n+1)+...+n(2n+1)

Answer 8

exactly - now factor out the (2n+1)

Answer 9

i think like this way:
i will assume that this is true until line n+1 (go downward from first line), i have: first line is true, 1+n+n = 2n + 1. so with line: n+1 we have: (n+1) + ... (n+1) at first triangle, at second triangle, we have: 1+ 2 + ... + n+ n+1, and at third: n+1 + n + ... + 1, sum last line, we still get: 2(n+1) + 1 + .... 2(n+1) + 1

Answer 10

\[\frac{n\left(n+1\right) \left(2n+1\right)}2\]What about the six?

Answer 11

that is correct - now you have to subtract the sum of the previous 2 triangles from this

Answer 12

but why?

Answer 13

what you are trying to find is the sum of the numbers in the 1st triangle

Answer 14

\[1^1+2^2+3^3+...\]

Answer 15

that should have started with \(1^2\)

Answer 16

it is a visual proof of this sum

Answer 17

oh I see it

Answer 18

It's an interesting proof - I suggest you try and do it yourself - you'll have greater pleasure from that :)

Answer 19

thanks, i will try my best.

Answer 20

let me know if you require a clue.

Answer 21

when you do "see" how to do it - it is quite remarkable!

Answer 22

@Ishaan94 - try turning your head as you view the triangles on the left...

Answer 23

this really is a "visual" solution - so don't think about any complex math here

Answer 24

OMG this is the best proof I have seen in my life so far

Answer 25

:) glad you "saw" it at last!

Answer 26

it is a very pleasing proof

Answer 27

it is. and thanks a lot asnaseer. :-)

Answer 28

yw :)

Answer 29

really beautiful xD