## Ishaan94 3 years ago Whom of you can explain it to me? http://jeremykun.files.wordpress.com/2011/06/triangle-proof.png

1. Ishaan94
2. Ishaan94

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

3. asnaseer

BTW: Nice site.

4. asnaseer

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

5. Ishaan94

No ._.

6. Ishaan94

wait

7. Ishaan94

Is it like this? (2n+1)+2(2n+1)+3(2n+1)+...+n(2n+1)

8. asnaseer

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

9. hosiduy

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

10. Ishaan94

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

11. asnaseer

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

12. Ishaan94

but why?

13. asnaseer

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

14. asnaseer

$1^1+2^2+3^3+...$

15. asnaseer

that should have started with $$1^2$$

16. asnaseer

it is a visual proof of this sum

17. Ishaan94

oh I see it

18. asnaseer

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

19. Ishaan94

thanks, i will try my best.

20. asnaseer

let me know if you require a clue.

21. asnaseer

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

22. asnaseer

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

23. asnaseer

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

24. Ishaan94

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

25. asnaseer

:) glad you "saw" it at last!

26. asnaseer

it is a very pleasing proof

27. Ishaan94

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

28. asnaseer

yw :)

29. rational

really beautiful xD