Question

If $p^{th}$ $q^{th}$ and $r^{th}$ term of an A.P. are $a,b$ and $c$ respectively, then $a(q-r)+b(r-p)+c(p-q)=?$

• A. $2$
• B. $1$
• C. $0$
• D. None of these

Answer 1

The correct option is C $0$

Let A be the first term of the A.P and D be the common difference of the A.P.
Given that
a= pth term
$\therefore a=A+(p-1)D$
b = qth term
$\therefore a=A+(q-1)D$
c= rth term
$\therefore a=A+(r-1)D$
L.H.S $=a(q-r)+b(r-p)+c(p-q)=aq-ar+br-bp+cp-cq$
$[(A+(p-1)D)(q-r)+(A+(q-1)D)(r-p)+(A+(r-1)D)(p-q)]$
$=[A((q-r)+(r-p)+(p-q))+D((p-1)(q-r)+(q-1)(r-p)+(r-1)(p-q))]$
$=[A\times 0+D(p(q-r)+q(r-p)+r(p-q)-(q-r)-(r-p)-(p-q))]$
$=[A\times 0+D\times 0]=0$