If A, B, C are the p^{th}, q^{th}, and r^{th} | vedclass

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(B) Let the first term be $a$ and the common ratio be $R$ of the $GP$.Then,$A = aR^{p-1}$,$B = aR^{q-1}$,and $C = aR^{r-1}$.Now,consider the expressio

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$1$

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(B) Let the first term be $a$ and the common ratio be $R$ of the $GP$.Then,$A = aR^{p-1}$,$B = aR^{q-1}$,and $C = aR^{r-1}$.Now,consider the expression $E = A^{q-r} \cdot B^{r-p} \cdot C^{p-q}$.Substituting the values:$E = (aR^{p-1})^{q-r} \cdot (aR^{q-1})^{r-p} \cdot (aR^{r-1})^{p-q}$$E = a^{(q-r) + (r-p) + (p-q)} \cdot R^{(p-1)(q-r) + (q-1)(r-p) + (r-1)(p-q)}$Since $(q-r) + (r-p) + (p-q) = 0$,the power of $a$ is $0$.For the power of $R$:$(pq - pr - q + r) + (qr - pq - r + p) + (rp - rq - p + q) = 0$Thus,$E = a^0 \cdot R^0 = 1 \cdot 1 = 1$.

If $A, B, C$ are the $p^{th}, q^{th},$ and $r^{th}$ terms of a $GP$ respectively,then $A^{q-r} \cdot B^{r-p} \cdot C^{p-q} =$

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