Докажите тождество: n(n+1)(n+2)+(n+1)=(n+1)^3.

\[n(n + 1)(n + 2) + (n + 1) = (n + 1)^{3}\]

\[(n + 1)\left( n(n + 2) + 1 \right) = (n + 1)^{3}\]

\[(n + 1)\left( n^{2} + 2n + 1 \right) = (n + 1)^{3}\]

\[(n + 1)(n + 1)^{2} = (n + 1)^{3}\]

\[(n + 1)^{3} = (n + 1)^{3}.\]

\[Что\ и\ требовалось\ доказать.\]