The Series of Madness

Algebra Level 5

A=n=150(xn+1xn)B=n=110(x5n2+1x5n2) \large { \begin{aligned} A&=& \sum_{n=1}^{50} \left(x^n+\frac1{x^n} \right) \\ B&=&\sum_{n=1}^{10} \left(x^{5n-2}+\frac1{x^{5n-2}} \right) \end{aligned}}

Given that x+1x=4x + \frac1x =4 , with AA and BB as described above, find the value of AB\dfrac AB.

This is one of my original Madness problems.
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