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Question Number 67836 by gunawan last updated on 01/Sep/19
If  determinant ((x^n ,x^(n+2) ,x^(n+3) ),(y^n ,y^(n+2) ,y^(n+3) ),(z^n ,z^(n+2) ,z^(n+3) ))  = (x−y)(y−z)(z−x)((1/x) + (1/y) + (1/z)),  then n equals
$$\mathrm{If}\:\begin{vmatrix}{{x}^{{n}} }&{{x}^{{n}+\mathrm{2}} }&{{x}^{{n}+\mathrm{3}} }\\{{y}^{{n}} }&{{y}^{{n}+\mathrm{2}} }&{{y}^{{n}+\mathrm{3}} }\\{{z}^{{n}} }&{{z}^{{n}+\mathrm{2}} }&{{z}^{{n}+\mathrm{3}} }\end{vmatrix} \\ $$$$=\:\left({x}−{y}\right)\left({y}−{z}\right)\left({z}−{x}\right)\left(\frac{\mathrm{1}}{{x}}\:+\:\frac{\mathrm{1}}{{y}}\:+\:\frac{\mathrm{1}}{{z}}\right), \\ $$$$\mathrm{then}\:{n}\:\mathrm{equals} \\ $$

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