If x, y, z are all positive and are the p th, q th and r th terms of a geometric progression respectively, then the value of the determinant | array lll x & p & 1 y & q & 1 z & r & 1 array | equals

Step-by-step solution

Let a and R be the first term and common ratio of a GP. array ll & T_p=a R^ p-1 =x & T_q=a R^ q-1 =y and & T_r=a R^ r-1 =z array and T_r=a R^ r-1 =z array rlrl & x & = a+(p-1) R & & y & = a+(q-1) R and & & z & = a+(r-1) R array aligned & | array lll x & p & 1 y & q & 1 z & r & 1 array |= | array lll a+(p-1) R & p & 1 a+(q-1) R & q & 1 a+(r-1) R & r & 1 array | & = | array lll a & p & 1 a & q & 1 a & r & 1 array |+ | array ccc (p-1) R & p & 1 (q-1) R & q & 1 (r-1) R & r & 1 array | & = a | array lll 1 & p & 1 1 & q