Summation formula for $x^2+x$

Question

Since I learned easier ways of calculating summations I've been curious as to how I could find formulas for as many equations as possible. I came across the equation $x^2+x$, I've spent quite some time on this problem and could not find a solution. If someone has maybe already done this or have any suggestions on how I could get the formula that would be greatly appreciated.

Example of another summation with a equation: $\sum\limits_{i=1}^n$ = $x^2$

Equation to solve this is $\frac{n(n+1)(2n+1)}{6}$

Answer 1

Extending what Git Gud said: $$\sum^n_{k=0}\left(k^2+k\right)=\sum^n_{k=0}k^2+\sum^n_{k=0}k=\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}=\frac{n^3}{3}+n^2+\frac{2n}{3}$$

Answer 2

Note that $$k^2+k=\frac{1}{3}\left((k+1)^3-k^3-1\right).$$ Thus our sum $\sum_{k=1}^n (k^2+k)$ is equal to $$\frac{1}{3}\left((2^3-1^3-1)+(3^3-2^3-1)+(4^3-3^3-1)+(5^3-4^3-1)+\cdots +((n+1)^3-n^3-1)\right).$$ Observe the nice almost total cancellations. We end up with $$\frac{1}{3}\left((n+1)^3-1^3-n\right).$$

Remarks: $1.$ Since we know $\sum_1^n k$, this gives a way to derive the formula for $\sum_1^n k^2$.

$2.$ The sums $\sum k(k+1)$, $\sum k(k+1)(k+2)$, $\sum k(k+1)(k+2)(k+3)$ and so on are nice, much nicer than $\sum k^2$, $\sum k^3$, $\sum k^4$ and so on.