<s>[INST] Squared Summation(Intermediate Step) I am studying Economics and are trying to get a firm grasp of summation rules and applications. Looking into the following relation,
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}$
The following "trick" is given below, to understand the above.
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$\sum_{k=1}^n[(k+1)^3-k^3]=n^3 + 3n^2+3n$
As I expand the left-handside of the equation for a given sequence $S=[1^2, 2^2, 3^3,..., n^3]$, the following leads to the right-hand side of the equation,
$(2^3 - 1^3) + (3^3 - 2^3) + (4^3 - 3^3) + .... + (n^3 - (n - 1)^3) + ((n+1)^3 - n^3) = (n + 1)^3 - 1^3 = n^3 + 3n^2 + 3n$
Understanding the intermediate steps, ie. the above expanding, Im struggling with the intuition so to speak. Which kind of mentality should I have had applied on the second equation, in order get to the right hand side, without expanding it?
Any help, with some mathematical explanation is highly appreciated. 
Thank you!
 [/INST] You need to know also that $1+2+...+n=\frac{n(n+1)}{2}$ and use your work:
$$n^3+3n^2+3n=3x+3\cdot\frac{n(n+1)}{2}+n,$$ which gives which you wish.
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