If pth, qth, rth terms of an AP are in GP whose commom ratio is k, then the root of the equation $(q-r) *x^2+(r-p) *x+(p-q) =0$ other than unity is

Using the relationship between the roots and coefficients for (q - r)x^2 + (r - p)x + (p - q) = 0, the product of the roots is (p - q)/(q - r). Since one root is unity, the other root is k. This is because substituting the arithmetic progression terms as A + (p-1)D into the common ratio formula k = (qth term)/(pth term) gives k = (A + (q-1)D)/(A + (p-1)D), and rearranging this proportionally demonstrates that (p - q)/(q - r) simplifies directly to k.

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