If x=\frac{6}{7-\frac{6}{7-\frac{6}{7-x}}}; x \gt 1, then what is the value of x^{2}-3x+2 equal to?

Explanation

Given the nested structure, if we assume the sequence generates x = \frac{6}{7-x}, then x(7-x) = 6 \implies x^2-7x+6=0, giving roots x=1 and x=6. Since x \gt 1, we take x=6. Substituting x=6 into the original finite nested fraction yields exactly 6. Then x^2-3x+2 = 6^2-3(6)+2 = 36-18+2 = 20.

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