If (1/a), (1/b), (1/c) are the p th , q th , r th terms respectively of an A.P. then the value of ab(p-q)+b c(q-r)+c a(r-p) is

Question

If (1/a), (1/b), (1/c) are the p th , q th , r th terms respectively of an A.P. then the value of ab(p-q)+b c(q-r)+c a(r-p) is (A) -1 (B) 2 (C) 0 (D) -2

Tardigrade Admin · Accepted Answer

Let x be the first term and y be the c.d. of corresponding A.P., then (1/a)=x+(p-1) y ...... (i) (1/ b )= x +( q -1) y ...... (ii) (1/ c )= x +( r -1) y ..... (iii) Multiplying (i), (ii) and (iii) respectively by abc (q - r), abc (r - p), abc (p - q) and then adding, we get, bc (q - r) + ca (r - p)+ ab (p - q) = 0

Q. If a1​,b1​,c1​ are the pth,qth,rth terms respectively of an A.P. then the value of ab(p−q)+bc(q−r)+ca(r−p) is

-1

2

0

-2

Let x be the first term and y be the c.d. of corresponding A.P., then a1​=x+(p−1)y......(i) b1​=x+(q−1)y......(ii) c1​=x+(r−1)y.....(iii) Multiplying (i), (ii) and (iii) respectively by abc(q−r),abc(r−p),abc(p−q) and then adding, we get, bc(q−r)+ca(r−p)+ab(p−q)=0

Q. If $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are the $p^{t h}, q^{t h}, r^{t h}$ terms respectively of an A.P. then the value of $ab (p - q) + b c (q - r) + c a (r - p)$ is