<s>[INST] Find all $x$ such that $2^x,2^{x^2}$ and $2^{x^3}$ form $3$ terms of an A.P. I know that if $a,b,c$ are in Arithmetic Progression, then $2b=a+c$, but in this case, I am unable to solve for $x$. Hints are appreciated. Thanks.
 [/INST] HINT:
$$2^x(1+2^{x^3-x})=2^{x^2+1}\iff 1+2^{x^3-x}=2^{x^2-x+1}$$
If $x^3-x>0,1+2^{x^3-x}>1$ is odd unlike $2^{x^2-x+1}$ as $x^2-x+1>0$ for real $x$
So, $x^3-x$ must be $0$
and consequently, $2=2^{x^2-x+1}\implies 1=x^2-x+1$
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