<s>[INST] Solve for $x^{11} + \frac{1}{x^{11}}$ if $x^3 + \frac{1}{x^3} = 18.$ Solve for $x^{11} + \frac{1}{x^{11}}$ if $x^3 + \frac{1}{x^3} = 18.$
I'm familiar with variants of this problem, especially those where the exponent in the given is a factor of the exponent in what we want to solve for, but this one stumped me. 
 [/INST] A little deus ex machina but this seems to be related to the golden ratio $\phi = \frac{1+\sqrt 5}{2}$, which is the solution of $x+\frac1x = 1$, and to the Fibonacci sequence $1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657, \ldots$  
In particular $x^3+\frac1{x^3} = 18$ has the solutions $x^3=5+8\phi$ and its reciprocal $-8\phi+13$, which requires $x=1+\phi$ or its reciprocal $-\phi+2$, and so $x^{11}=10946+17711\phi$ or its reciprocal $-17711\phi + 28657$ which gives $$x^{11} + \frac{1}{x^{11}} = 10946+17711\phi -17711\phi + 28657 = 39603.$$
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