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

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(B) Let the first term of the $G.P.$ be $A$ and the common ratio be $R$.The $p^{th}$ term is $a = A R^{p-1}$  ---$(i)$The $q^{th}$ term is $b = A R^{q

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(B) Let the first term of the $G.P.$ be $A$ and the common ratio be $R$.The $p^{th}$ term is $a = A R^{p-1}$  ---$(i)$The $q^{th}$ term is $b = A R^{q-1}$  ---(ii)The $r^{th}$ term is $c = A R^{r-1}$  ---(iii)Now,consider the expression $E = a^{q - r} b^{r - p} c^{p - q}$.Substituting the values of $a, b,$ and $c$:$E = (A R^{p-1})^{q-r} (A R^{q-1})^{r-p} (A R^{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)}$Calculating the exponent of $A$:$(q-r) + (r-p) + (p-q) = 0$Calculating the exponent of $R$:$(pq - pr - q + r) + (qr - qp - r + p) + (rp - rq - p + q) = 0$Thus,$E = A^0 \cdot R^0 = 1 \cdot 1 = 1$.

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of a $G.P.$ are $a$,$b$,and $c$ respectively,then $a^{q - r} b^{r - p} c^{p - q}$ is equal to

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