Sum of the number sequence that consists of multiplications of neighbor numbers

Question

Say there is sequnce of number $A(i) = i * (i+1)$, i.e. like 1*2, 2*3, 3*4 and so on. Is there any formula that allows to compute sum of the first N such numbers?

Answer 1

Use that $$ k(k+1)(k+2)-(k-1)k(k+1)=k(k+1)·((k+2)-(k-1))=3·k(k+1) $$ to find a representation as telescoping sum.

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This can be rewritten as $\binom{k+2}{3}-\binom{k+1}{3}=\binom{k+1}{2}$.

One can use the identity of Pascals triangle in general in an analogous way, as $$ \binom{k+m-1}m-\binom{k+m-2}m=\binom{k+m-2}{m-1}=\frac{k(k+1)···(k+m-2)}{(m-1)!} $$