2011/082) The equation x³ + px² + qx + r = 0 (where p, q, r are non zero) has roots α, β, γ such that 1/α , 1/β, 1/γ...?

are consecutive terms in an arithmetic sequence, show that β = -3r / q

f(x) = x^3+px^2+qx+r = 0 has the roots α, β, γ

so f(1/x) has the roots 1/α , 1/β, 1/γ. Which are in AP

f(1/x) = 1/x^3+p/x^2+q/x+ r = 0

or rx^3+qx^2+px+1 = 0

sum of roots = - q/r = 3/ β or β = -3r/q (sum of 3 terms of AP = 3 * middle term)

proved