<s>[INST] Simplify the following $(1\cdot2)+(2\cdot3)+(3\cdot4)+\dots+(n\cdot(n+1))$ Simplify the following $(1\cdot2)+(2\cdot3)+(3\cdot4)+\dots+(n\cdot(n+1))$.
How do i do this by using these identities.
$C^r_r + C^{r+1}_r + C^{r+2}_r + ....+C^{n}_r = C^{n+1}_{r+1}$
Or
$C^r_0 + C^{r+1}_1 + C^{r+2}_2 + ....+C^{r+k}_k = C^{r+k+1}_{k}$
 [/INST] Hint:
$$(1\times2)+(2\times3)+(3\times4)+\dots(n\times(n+1)) =\sum_{k=1}^nk(k+1) \\
\begin{align}
& =2\sum_{k=1}^n\frac{k(k+1)}2 \\
& =2\sum_{k=1}^nC_2^{k+1} \\
\end{align}$$
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