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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.

 

 

 

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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
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