<s>[INST] For every natural number n, prove that the vector $x = (n, n + 1, n (n + 1)) ∈ \Bbb R^3$ has as a norm a natural number, that is, $\|x\| \in \Bbb N$. 
For every natural number $n$, prove that the vector $x = (n, n + 1, n (n + 1))$ ∈ $\mathbb{R}^3$ has as a norm a natural number, that is, $\|x\| \in \mathbb{N}$.

I made the internal product of the vector $x$ which gave $n^4 + 3n^3 + n^2$. Now I can't prove by induction. Does anyone have an idea?
 [/INST] \begin{align*}
\sqrt{n^2+(n+1)^2+n^2(n+1)^2}&=\sqrt{(n^2+1)(n+1)^2+n^2}\\
&=\sqrt{(n^2+1+2n)(n+1)^2+n^2-2n(n+1)^2}\\
&=\sqrt{(n+1)^4-2n(n+1)^2+n^2}\\
&=\sqrt{[(n+1)^2-n]^2}\\
&=(n+1)^2-n\\
&=n^2+n+1. 
\end{align*}
This is obviously a natural number for $n\in\mathbb{N}$.
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