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Question

Find the value of the determinant ∣ ∣bccaabpqr111∣ ∣ where a, b and c are respectively the pth, qth and rth terms of a harmonic progression.

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Solution

Given a,b and c are respectively the pth,qth and rth terms of a harmonic progression.
a=1A+(p1)D,b=1A+(q1)D and c=1A+(r1)D
where A,D are first term and common difference of corresponding Arithmetic progression.
∣ ∣bccaabpqr111∣ ∣=abc∣ ∣1/a1/b1/cpqr111∣ ∣
=abc∣ ∣A+(p1)DA+(q1)DA+(r1)Dpqr111∣ ∣
applying R1R1AR3 gives
=abcD∣ ∣p1q1r1pqr111∣ ∣
applying R1+R3 gives
=abcD∣ ∣pqrpqr111∣ ∣=0
∣ ∣bccaabpqr111∣ ∣=0

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