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 be $A$ and the common difference be $D$.The $p^{th}$ term is $a = A + (p - 1)D$  $(i)$The $q^{th}

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$0$

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(C) Let the first term of the Arithmetic Progression be $A$ and the common difference be $D$.The $p^{th}$ term is $a = A + (p - 1)D$  $(i)$The $q^{th}$ term is $b = A + (q - 1)D$  $(ii)$The $r^{th}$ term is $c = A + (r - 1)D$  $(iii)$Now,consider the expression $E = a(q - r) + b(r - p) + c(p - q)$.Substituting the values of $a, b, c$:$E = [A + (p - 1)D](q - r) + [A + (q - 1)D](r - p) + [A + (r - 1)D](p - q)$$E = A(q - r + r - p + p - q) + D[(p - 1)(q - r) + (q - 1)(r - p) + (r - 1)(p - q)]$$E = A(0) + D[pq - pr - q + r + qr - qp - r + p + rp - rq - p + q]$$E = 0 + D[0] = 0$.

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an Arithmetic Progression are $a$,$b$,and $c$ respectively,then $[a(q - r) + b(r - p) + c(p - q)] = ?$

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