Solve the following


If pthqth and rth terms of a G.P. are xyz respectively, then write the value of xq − r yr − p zp − q.


Let us take a G.P. whose first term is and common ratio is R.

According to the question, we have:

$A R^{p-1}=x$

$A R^{q-1}=y$

$A R^{r-1}=z$

$\therefore x^{q-r} y^{r-p} z^{p-q}$

$=A^{q-r} \times R^{(p-1)(q-r)} \times A^{r-p} \times R^{(q-1)(r-p)} \times A^{p-q} \times R^{(r-1)(p-q)}$

$=A^{q-r+r-p+p-q} \times R^{(p r-p r-q+r)+(r q-r+p-p q)+(p r-p-q r+q)}$

$=A^{0} \times R^{0}$


$\therefore x^{q-r} y^{r-p} z^{p-q}=1$

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