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

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

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(C) Let the first term be $A$ and the common difference be $D$ for the arithmetic progression.The $p^{th}$ term is $A + (p - 1)D = a$ $(i)$The $q^{th}$ term is $A + (q - 1)D = b$ $(ii)$The $r^{th}$ term is $A + (r - 1)D = c$ $(iii)$Subtracting $(ii)$ from $(i)$,$(i)$ from $(iii)$,and $(iii)$ from $(ii)$ gives:$a - b = (p - q)D$$b - c = (q - r)D$$c - a = (r - p)D$Now,consider the expression $E = a(q - r) + b(r - p) + c(p - q)$.Substituting the values of $(q - r)$,$(r - p)$,and $(p - q)$ from the differences above:$E = a\left(\frac{b - c}{D}\right) + b\left(\frac{c - a}{D}\right) + c\left(\frac{a - b}{D}\right)$$E = \frac{1}{D} [a(b - c) + b(c - a) + c(a - b)]$$E = \frac{1}{D} [ab - ac + bc - ab + ca - bc]$$E = \frac{1}{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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