If Pth, Qth and Rth Terms of a G.P. Re X, Y, Z Respectively, Then Write the Value of Xq − R Yr − Pzp − Q. - Mathematics

If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.

Solution

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

$\text{ According to the question, we have }:$
$A R^{p - 1} = x$
$A R^{q - 1} = y$
$A R^{r - 1} = z$
$\therefore x^{q - r} y^{r - p} z^{p - q}$
$= A^{q - r} \times R^\left( p - 1 \right)\left( q - r \right) \times A^{r - p} \times R^\left( q - 1 \right)\left( r - p \right) \times A^{p - q} \times R^\left( r - 1 \right)\left( p - q \right)$
$= A^{q - r + r - p + p - q} \times R^\left( pr - pr - q + r \right) + \left( rq - r + p - pq \right) + \left( pr - p - qr + q \right)$
$= A^0 \times R^0$
$= 1$
$\therefore x^{q - r} y^{r - p} z^{p - q} = 1$


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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 20 Geometric Progression
Q 5 | Page 56