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

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

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$0$

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(C) Let the first term of the arithmetic progression $(A.P.)$ be $A$ and the common difference be $D$.The $p^{th}$ term is given by $a = A + (p - 1)D$ .....$(i)$The $q^{th}$ term is given by $b = A + (q - 1)D$ .....$(ii)$The $r^{th}$ term is given by $c = A + (r - 1)D$ .....$(iii)$We need to evaluate the expression $E = a(q - r) + b(r - p) + c(p - q)$.Substituting the values of $a, b,$ and $c$:$E = [A + (p - 1)D](q - r) + [A + (q - 1)D](r - p) + [A + (r - 1)D](p - q)$Expanding the terms:$E = A(q - r + r - p + p - q) + D[(p - 1)(q - r) + (q - 1)(r - p) + (r - 1)(p - q)]$Since $(q - r + r - p + p - q) = 0$,the first part becomes $A(0) = 0$.For the second part:$(p - 1)(q - r) = pq - pr - q + r$$(q - 1)(r - p) = qr - qp - r + p$$(r - 1)(p - q) = rp - rq - p + q$Summing these:$(pq - qp) + (-pr + rp) + (-qr + rq) + (-q + q) + (-r + r) + (-p + p) = 0$Thus,$E = A(0) + D(0) = 0$.

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an arithmetic sequence are $a$,$b$,and $c$ respectively,then the value of $[a(q - r) + b(r - p) + c(p - q)]$ is:

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