Question

Find the value of the determinant `|b cc a a b p q r1 1 1|,w h e r ea ,b ,a n dc` are respectively, the pth, qth, and rth terms of a harmonic progression.

Answer 1

Given that a,b and c pth qth rth terms respectively of a H.P.
So `(1)/(a),(1)/(b),(1)/(c) ` are pth qth rth terms of an A.P.
Therefore ,
`(1)/(a) =A +(p-1) D,`
`(1)/(b) =A +(q-1)D,`
`(1)/(C) =A +(r-1)D.`
where a is first terms and D is common difference of A.P... Given determinant is
`Delta =|{:(bc,,ca,,ab),(p,,q,,r),(1,,1,,1):}|`
` = abc |{:((1)/(a),,(1)/(b),,(1)/(c)),(p,,q,,r),(1,,1,,1):}|`
`=abc |{:(A+(p-1)D,,A+(q-1)D,,A+(r-1)D),(p,,q,,r),(1,,1,,1):}|`
Applying `R_(1) to R_(2) -(A-D) R_(3) -D R_(2)` we get
`Delta =abc |{:(0,,0,,0),(p,,q,,r),(1,,1,,1):}|=0`