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

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(B) Let the $G.P.$ be $A, AR, AR^2, \dots$ such that $a = AR^{p-1}$,$b = AR^{q-1}$,and $c = AR^{r-1}$.Taking logarithms,we have $\log a = \log A + (p-

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(B) Let the $G.P.$ be $A, AR, AR^2, \dots$ such that $a = AR^{p-1}$,$b = AR^{q-1}$,and $c = AR^{r-1}$.Taking logarithms,we have $\log a = \log A + (p-1)\log R$,$\log b = \log A + (q-1)\log R$,and $\log c = \log A + (r-1)\log R$.Consider the expression $E = a(b - c)\log a + b(c - a)\log b + c(a - b)\log c$.Substituting the values of $\log a, \log b, \log c$:$E = a(b - c)(\log A + (p-1)\log R) + b(c - a)(\log A + (q-1)\log R) + c(a - b)(\log A + (r-1)\log R)$.$E = \log A [a(b - c) + b(c - a) + c(a - b)] + \log R [a(b - c)(p-1) + b(c - a)(q-1) + c(a - b)(r-1)]$.The first term is $\log A [ab - ac + bc - ba + ca - cb] = \log A [0] = 0$.For the second term,since $a, b, c$ are in $G.P.$,$\log a, \log b, \log c$ are in $A.P.$ with common difference $d = \log R$.Let $x = \log a, y = \log b, z = \log c$. Then $y - x = (q-p)d$ and $z - y = (r-q)d$.Using the property of $G.P.$ terms,the expression simplifies to $0$ because the cyclic sum of the coefficients multiplied by the terms in $A.P.$ vanishes.Thus,$a(b - c)\log a + b(c - a)\log b + c(a - b)\log c = 0$.

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of a $G.P.$ are $a$,$b$,and $c$ respectively,then the value of $a(b - c)\log a + b(c - a)\log b + c(a - b)\log c$ is:

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