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

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(B) Let the first term of the geometric progression be $A$ and the common ratio be $R$. Given that the $p^{th}$,$q^{th}$,and $r^{th}$ terms are $a, b,

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$1$

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(B) Let the first term of the geometric progression be $A$ and the common ratio be $R$. Given that the $p^{th}$,$q^{th}$,and $r^{th}$ terms are $a, b, c$ respectively,we have: $a = AR^{p-1}$,$b = AR^{q-1}$,and $c = AR^{r-1}$. Now,consider the expression $a^{q-r} \cdot b^{r-p} \cdot c^{p-q}$: $= (AR^{p-1})^{q-r} \cdot (AR^{q-1})^{r-p} \cdot (AR^{r-1})^{p-q}$ $= A^{(q-r) + (r-p) + (p-q)} \cdot R^{(p-1)(q-r) + (q-1)(r-p) + (r-1)(p-q)}$ $= A^0 \cdot R^{(pq - pr - q + r) + (qr - qp - r + p) + (rp - rq - p + q)}$ $= 1 \cdot R^0 = 1$.

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of a geometric progression are $a, b, c$ respectively,then $a^{q-r} \cdot b^{r-p} \cdot c^{p-q} = \dots\dots$

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