The (p+q)th term and (p-q)th terms of a G.P. are a and b respectively. Find the pth term.

Answer

To solve the problem of finding the pth term of a geometric progression (G.P.) given that the (p+q)th term is \( a \) and the (p-q)th term is \( b \), we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Terms of G.P.**:
The nth term of a G.P. can be expressed as:
\[ T_n = A_1 \cdot R^{n-1} \]
where \( A_1 \) is the first term and \( R \) is the common ratio.

2. **Setting Up the Equations**:
From the problem, we know:
- The (p+q)th term is \( a \):
\[ T_{p+q} = A_1 \cdot R^{(p+q)-1} = a \]
- The (p-q)th term is \( b \):
\[ T_{p-q} = A_1 \cdot R^{(p-q)-1} = b \]

3. **Writing the Equations**:
We can write these two equations as:
\[ A_1 \cdot R^{p+q-1} = a \quad \text{(1)} \]
\[ A_1 \cdot R^{p-q-1} = b \quad \text{(2)} \]

4. **Multiplying the Two Equations**:
Now, we multiply equations (1) and (2):
\[ (A_1 \cdot R^{p+q-1}) \cdot (A_1 \cdot R^{p-q-1}) = a \cdot b \]
This simplifies to:
\[ A_1^2 \cdot R^{(p+q-1) + (p-q-1)} = ab \]
Simplifying the exponent:
\[ A_1^2 \cdot R^{2p - 2} = ab \]
or
\[ A_1^2 \cdot R^{2(p-1)} = ab \quad \text{(3)} \]

5. **Finding the pth Term**:
The pth term can be expressed as:
\[ T_p = A_1 \cdot R^{p-1} \]
To express this in terms of \( ab \), we can take the square root of equation (3):
\[ A_1 \cdot R^{p-1} = \sqrt{ab} \]
Therefore, we find:
\[ T_p = \sqrt{ab} \]

### Final Answer:
The pth term \( T_p \) of the G.P. is:
\[ T_p = \sqrt{ab} \]