If the p^{th},q^{th},and r^{th} terms of an A | vedclass

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(C) Let $A$ be the first term and $D$ be the common difference of the $A.P.$The $p^{th}$ term is $a = A + (p-1)D$ $...(1)$The $q^{th}$ term is $b = A

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$0$

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(C) Let $A$ be the first term and $D$ be the common difference of the $A.P.$The $p^{th}$ term is $a = A + (p-1)D$ $...(1)$The $q^{th}$ term is $b = A + (q-1)D$ $...(2)$The $r^{th}$ term is $c = A + (r-1)D$ $...(3)$Now,substitute these values into the expression $a(q-r) + b(r-p) + c(p-q)$:$= [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(0) + D[pq - pr - q + r + qr - qp - r + p + rp - rq - p + q]$$= 0 + D[0] = 0$Thus,the value of the expression is $0$.

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an $A.P.$ are $a, b,$ and $c$ respectively,find the value of $a(q-r) + b(r-p) + c(p-q)$.

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