How do you proove the formula for the triangle numbers?

n(n+1)/2
that's the formula. but i dont no how to proove it. can u help me? please

Sequence : 1, 3, 6, 10, 15, 21..

Answers:
well, lets say that the sum of the first n natural numbers is S
1+2+3+.+n=s
it is also true that
n+(n-1)+(n-2)+.+1=s
add the two equations together and you get
2s=(n+1)+(n+1)+(n+1)+(n+1)+.
on and on until you have n terms of (n+1) which can be written
2s=n(n+1)
or
s=n(n+1)/2

induction is great, but this makes more sense to me
You are trying to prove that
1 + 2 + 3 + . + n = n(n+1)/2
Use induction.
1) Show true for n = 1
1 = 1(1+1)/2
1 = 2/2
1 = 1
Thus, the formula is true for n = 1.
2) Assume the formula is true for n = k
1 + 2 + . + k = k(k+1)/2
3) Show that it is true for n = k+1
We need to show that
1 + 2 + . + k + (k+1) = (k+1)(k+1+1)/2
or
1 + 2 + . + k + (k+1) = (k+1)(k+2)/2
Use the assumption in step 2.
1 + 2 + . + k + (k+1) becomes
k(k+1)/2 + (k+1)
= (k^2 + k)/2 + 2(k+1)/2
= (k^2 + k + 2k +2)/2
= (k^2 + 3k + 2)/2
= (k+1)(k+2)/2
QED
You already have a sequence. You must prove that the statement P(1) [in this case n(n+1)/2, where n=1] is true.
Then if you assume that the statement P(k) is also true (for any arbitrary integer k), you must show that the statement P(k+1) is also true.
This is called mathematical induction. A google search for this will provide you with more information, but here's one link to give you an example: -

http://www.cut-the-knot.org/induction.sh.

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