<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] Note that $2\binom{k}{2}=(k-1)k$. Hence
$$(1\cdot2)+(2\cdot3)+(3\cdot4)+\dots+(n\cdot(n+1))=\sum_{k=2}^{n+1}2\binom{k}{2}=2\binom{n+2}{3}.$$
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