Using properties of determinants prove that:
where α, β, γ are in AP.
Given that α, β, γ are in an AP, which means 2β=α+γ
Operating R3→R3-2R2+R1
[we know that 2β=α+γ]
Operating R1→R1-R3, R2→R �2-R3
[By the properties of determinants, we know that if all the elements of a row or column is 0, then the value of the determinant is also 0]
=0
Hence proved
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