<s>[INST] Prove $x^n + x < (x^n)x$ using induction. I need to prove $x^n$ + x < $x^n\cdot$ x,   n $\in$ N, x $\in$ R>2  using induction. I started by 
$x^n$ + x + (x^(n+1)+x)  <  ($x^n\cdot$ x) + (x^(n+1)+x)
I simplified to this:
< 2x^(n+1) + x
I am stuck here. All help is appreciated.
 [/INST] Let $P(n): x^n+x< x^{n+1}, n \in \mathbb{N}, x > 2$.
Check $P(1)$ is true. $x^1+x = 2x < x^2 \iff x(x-2) > 0$ is true since $x > 2$.
Assume $P(n)$ is true, i.e. $x^n + x < x^{n+1}$, prove $P(n+1)$ is true.
$x^{n+1} + x = x\cdot x^n + x < x(x^{n+1}-x) + x = x^{n+2} - x^2 + x < x^{n+2} - x^2 + x^2 = x^{n+2}$ since $x^2 > x$. Thus $P(n+1)$ is true, and by MPI the statement is true.
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