If ( a > 0, ~b > 0, c > 0 ) are respectively the ( pth , q ) th, rth terms of GP., then the value of the determinant ( | array lll a & p & 1 b & q & 1 c & r & 1 array | ) is

Step-by-step solution

Let (A ) be the 1 st term and ( R ) the common ratio of G.P., then ( aligned & a=T_p=A R^ p-1 & a = A +( p -1) R aligned ) Similarly, ( b = A +( q -1) R ) and ( c = A +( r -1) R ) ( = | array lll A+(p-1) R & p & 1 A+(q-1) R & q & 1 A+(r-1) R & r & 1 array | ) Split into two determinants and in the first take ( A ) common and in the second take ( R ) common ( = A | array lll 1 & p & 1 1 & q & 1 1 & r & 1 array |+ R | array lll p-1 & p & 1 q-1 & q & 1 r-1 & r & 1 array | ) Apply ( C ₁ C ₂ C ₃ ) in the second ( =0+ R