## Ishaan94 Group Title Whom of you can explain it to me? http://jeremykun.files.wordpress.com/2011/06/triangle-proof.png 2 years ago 2 years ago

1. Ishaan94 Group Title
2. Ishaan94 Group Title

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 Group Title

BTW: Nice site.

4. asnaseer Group Title

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

5. Ishaan94 Group Title

No ._.

6. Ishaan94 Group Title

wait

7. Ishaan94 Group Title

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

8. asnaseer Group Title

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

9. hosiduy Group Title

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 Group Title

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

11. asnaseer Group Title

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

12. Ishaan94 Group Title

but why?

13. asnaseer Group Title

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

14. asnaseer Group Title

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

15. asnaseer Group Title

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

16. asnaseer Group Title

it is a visual proof of this sum

17. Ishaan94 Group Title

oh I see it

18. asnaseer Group Title

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

19. Ishaan94 Group Title

thanks, i will try my best.

20. asnaseer Group Title

let me know if you require a clue.

21. asnaseer Group Title

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

22. asnaseer Group Title

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

23. asnaseer Group Title

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

24. Ishaan94 Group Title

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

25. asnaseer Group Title

:) glad you "saw" it at last!

26. asnaseer Group Title

it is a very pleasing proof

27. Ishaan94 Group Title

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

28. asnaseer Group Title

yw :)

29. rational Group Title

really beautiful xD