Here's the question you clicked on:
across
Let's go over the derivation of\[\sum_{i=1}^{n}i^2=\frac{n(n+1)(2n+1)}{6}\]
This could follow from Gauss' assertion that\[\sum_{i=1}^{n}i=\frac{n(n+1)}{2}\]
i don't remember the details. but i'm pretty sure u can find this in Spivaks' Calculus
It's easy to prove this by induction with your eyes closed, but I'm wondering how it's derived.
Can I provide the link or not here ??
@across can I provide here the link or I have to derive here full??
i think this was done in physics section ...
This is nothing but sum of squares of first n natural numbers..
yes i know the proof is easy. i suggested spivak's book because i remember seeing the derivation there
i would prefer geometrical visualization instead ... i remember seeing one is mit ocw single variable calculus. the volume of pyramid ...
Links are more appreciated than derivations here. ^^
http://www.ilovemaths.com/3sequence.asp May be it will help you.
http://www.9math.com/book/sum-squares-first-n-natural-numbers Or you can check this also..
http://www.trans4mind.com/personal_development/mathematics/series/sumNaturalSquares.htm#general_term
difference of power is very power technique ... it can be used to find the sums of n^3 and n^4 also ....
There is a very beautiful visual proof of this that @Ishaan94 came across when he asked this question: http://openstudy.com/updates/5002f7dae4b0848ddd66eea4
very interesting proof indeed !! \[ 3 \left ( \sum_{n=1}^\infty n^2 \right ) = (1 + 2 +3 +... + n)(2n+1)\]