Solution
$$a+\ \frac{\ 1}{a}+2=0$$
$$a^2+1+2a=0$$
$$\left(a+1\right)^2=0$$
$$=$$> $$a=-1$$
Therefore, $$a^{37}-\ \frac{\ 1}{a^{100}}=\left(-1\right)^{37}-\ \frac{\ 1}{\left(-1\right)^{100}}$$
$$=-1-1$$
$$=-2$$
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