Question

In a G.P. of $ (m + n)^{\text{th}} $ term is $ P $ and the $ (m - n)^{\text{th}} $ term is $ q $, then its $ m^{\text{th}} $ term is

• A. 0
• B. \( Pq \)
• C. \( \sqrt{Pq} \)
• D. \( \frac{1}{2}(P + q) \)

Hint:

For geometric progressions, use the general term formula \( T_n = ar^{n-1} \) and apply the given information to solve for unknown terms.

Answer 1

The Correct Option is C

Step 1: Understand the concept of G.P.
(Geometric Progression).
In a geometric progression (G.P.), the \( n^{\text{th}} \) term is given by: \[ T_n = ar^{n-1} \] where \( a \) is the first term and \( r \) is the common ratio.
Step 2: Use the given information.
Let the first term be \( a \) and the common ratio be \( r \).
We are given that the \( (m + n)^{\text{th}} \) term is \( P \), and the \( (m - n)^{\text{th}} \) term is \( q \).
Using the formula for the \( n^{\text{th}} \) term of a G.P., we get the following equations: \[ T_{m+n} = ar^{m+n-1} = P \] \[ T_{m-n} = ar^{m-n-1} = q \]
Step 3: Find the \( m^{\text{th}} \) term.
The \( m^{\text{th}} \) term is: \[ T_m = ar^{m-1} \] Now, using the given equations for \( P \) and \( q \), we can solve for \( T_m \): \[ T_m = \sqrt{Pq} \]
Step 4: Conclusion.
The \( m^{\text{th}} \) term is \( \sqrt{Pq} \), which is option (c).