If (2p)th term of a G.P is `q^2` and (2q)th term is `p^2` then (p+q)th term is

Options

A. pq
B. `(1)/(2) p^(2) q^(2)`
C. `p^(2) q^(2)`
D. `(1)/(4) p^(3) q^(3)`

Answer

To solve the problem step by step, we will use the properties of a geometric progression (G.P.).

### Step-by-Step Solution:

1. **Understanding the Terms of G.P.**:
- The \( n \)th term of a G.P. can be expressed as:
\[ T_n = a \cdot r^{n-1} \]
where \( a \) is the first term and \( r \) is the common ratio.

2. **Expressing Given Terms**:
- We know that the \( (2p) \)th term is \( q^2 \):
\[ T_{2p} = a \cdot r^{2p-1} = q^2 \quad (1) \]
- Similarly, the \( (2q) \)th term is \( p^2 \):
\[ T_{2q} = a \cdot r^{2q-1} = p^2 \quad (2) \]

3. **Multiplying the Two Equations**:
- Multiply equations (1) and (2):
\[ (a \cdot r^{2p-1}) \cdot (a \cdot r^{2q-1}) = q^2 \cdot p^2 \]
- This simplifies to:
\[ a^2 \cdot r^{(2p-1) + (2q-1)} = p^2 q^2 \]
- Which can be rewritten as:
\[ a^2 \cdot r^{2p + 2q - 2} = p^2 q^2 \quad (3) \]

4. **Finding the \( (p+q) \)th Term**:
- The \( (p+q) \)th term can be expressed as:
\[ T_{p+q} = a \cdot r^{(p+q)-1} \]

5. **Relating the Terms**:
- From equation (3), we can express \( a \cdot r^{p+q-1} \):
\[ a \cdot r^{p+q-1} = \sqrt{p^2 q^2} = pq \]
- Therefore, we have:
\[ T_{p+q} = pq \]

### Final Answer:
The \( (p+q) \)th term of the G.P. is:
\[ \boxed{pq} \]