Given a,b and c are respectively the pth,qth and rth terms of a harmonic progression.
⇒a=1A+(p−1)D,b=1A+(q−1)D and c=1A+(r−1)D
where A,D are first term and common difference of corresponding Arithmetic progression.
∣∣
∣∣bccaabpqr111∣∣
∣∣=abc∣∣
∣∣1/a1/b1/cpqr111∣∣
∣∣
=abc∣∣
∣∣A+(p−1)DA+(q−1)DA+(r−1)Dpqr111∣∣
∣∣
applying R1→R1−AR3 gives
=abcD∣∣
∣∣p−1q−1r−1pqr111∣∣
∣∣
applying R1+R3 gives
=abcD∣∣
∣∣pqrpqr111∣∣
∣∣=0
∴∣∣
∣∣bccaabpqr111∣∣
∣∣=0