If a₁, a₂, a₃, 5,4, a₆, a₇, a₈, a₉ are in H.P., then the value of the determinant | array ccc a₁ & a₂ & a₃ 5 & 4 & a₆ a₇ & a₈ & a₉ array | can be expressed in the lowest form as p q , find (p+q) .

Step-by-step solution

We have D= | array ccc a₁ & a₂ & a₃ 5 & 4 & a₆ a₇ & a₈ & a₉ array | Since a_n= 20 n ; d= 1 20 Hence D= | array ccc 20 & 20 2 & 20 3 20 4 & 20 5 & 20 6 20 7 & 20 8 & 20 9 array |= (20)^3 4 7 | array ccc 1 & 1 2 & 1 3 1 & 4 5 & 2 3 1 & 7 8 & 7 9 array |R₁ R₁-R₂ and R₂ R₂-R₃= (20)^3 4 7 | array ccc 0 & -3 10 & -1 3 0 & -3 40 & -1 9 1 & 7 8 & 7 9 array |= 50 21 = p q aligned & p=50, q=21 & (p+q)=71 Ans. aligned