Question

If p^th, q^th and r^th terms of a G.P. re x, y, z respectively, then write the value of x^q − r y^r − pz^p − q.

 

 

 

Answer 1

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

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