## apple_pi 2 years ago Sum of first n triangle numbers

1. apple_pi

How do we find this?

2. cwrw238

1,3,6,10.. is this the series?

3. apple_pi

yeah, the one generated by n(n+1)/2

4. cwrw238

right - i remember that now but i don't recall the sum formula

5. cwrw238

i can only suggest googling it

6. UnkleRhaukus

|dw:1345110210994:dw|

7. cwrw238
8. cwrw238

@UnkleRhaukus - good drawing

9. apple_pi

Yeah, but that's only proving what we get, not how to get there

10. cwrw238

yes - i see what you mean - it seems that its a guess, which is then proved by induction

11. nightwill

$S_n=\Sigma \frac{n(n+1)}{2} = \frac{1}{2} \Sigma (n^2+n) = \frac{1}{2}(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2})$ $=\frac{1}{12}(n(n+1)(2n+1)+3n(n+1)) = \frac{1}{12} (n(n+1)(2n+1+3))$ $= \frac{1}{12}(n(n+1)(2n+4)) =\frac{2}{12}(n(n+1)(n+2)) = \frac{n(n+1)(n+2)}{6}$

12. apple_pi

How did you get the sum of n^2?