Determine the $\det(A)$ by induction, where:
$$A = \begin{pmatrix} 1 & x & x & . & . & . & x \\ x & 2 & x & . & . & . & x \\ x & x & 3 & . & . & . & x \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ x & x & x & . & . & . & n \\ \end{pmatrix}$$
Don't really know where to start besides computing the first few: $$-x^2+2\\ 2 x^{3} - 6 x^{2} + 6\\ - 3 x^{4} + 20 x^{3} - 35 x^{2} + 24\\ 4 x^{5} - 45 x^{4} + 170 x^{3} - 225 x^{2} + 120\\ - 5 x^{6} + 84 x^{5} - 525 x^{4} + 1470 x^{3} - 1624 x^{2} + 720\\ 6 x^{7} - 140 x^{6} + 1288 x^{5} - 5880 x^{4} + 13538 x^{3} - 13132 x^{2} + 5040\\ - 7 x^{8} + 216 x^{7} - 2730 x^{6} + 18144 x^{5} - 67347 x^{4} + 134568 x^{3} - 118124 x^{2} + 40320\\ 8 x^{9} - 315 x^{8} + 5220 x^{7} - 47250 x^{6} + 253092 x^{5} - 807975 x^{4} + 1447360 x^{3} - 1172700 x^{2} + 362880$$
Obvious pattern for the first and last coefficient of each polynomial, can't spot the others.
Do pivotal condensation. The determinant is$$\det( \text{ diag }(2-x^2, ... ,n-x^2))=\prod_{i=2}^n(i-x^2).$$