If 3rd, 8th and 13th terms of a G.P are p ,q and r respectively, then which one of the following is correct? (i) `q^(2)=pr` (ii) `r^(2)=pq` (iii) `pqr=1` (iv) `2q=p+r`

Options

A. `q^(2)=pr`
B. `r^(2)=pq`
C. `pqr=1`
D. `2q=p+r`

Answer

To solve the problem, we need to analyze the terms of the geometric progression (G.P.) given in the question.

### Step-by-Step Solution:

1. **Identify the Terms of the G.P.**:
- The 3rd term of the G.P. is given as \( p \).
- The 8th term of the G.P. is given as \( q \).
- The 13th term of the G.P. is given as \( r \).

2. **Use the Formula for the n-th Term of a 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.

3. **Write the Equations for Each Term**:
- For the 3rd term:
\[ p = a \cdot r^{3-1} = a \cdot r^2 \quad \text{(1)} \]
- For the 8th term:
\[ q = a \cdot r^{8-1} = a \cdot r^7 \quad \text{(2)} \]
- For the 13th term:
\[ r = a \cdot r^{13-1} = a \cdot r^{12} \quad \text{(3)} \]

4. **Square the Equation for the 8th Term**:
- From equation (2):
\[ q^2 = (a \cdot r^7)^2 = a^2 \cdot r^{14} \quad \text{(4)} \]

5. **Multiply the Equations for the 3rd and 13th Terms**:
- From equations (1) and (3):
\[ p \cdot r = (a \cdot r^2) \cdot (a \cdot r^{12}) = a^2 \cdot r^{14} \quad \text{(5)} \]

6. **Compare Equations (4) and (5)**:
- From (4) and (5), we have:
\[ q^2 = p \cdot r \]

7. **Conclusion**:
- Therefore, the correct relationship among the terms is:
\[ q^2 = p \cdot r \]
- This corresponds to option (i).

### Final Answer:
The correct option is (i) \( q^2 = pr \).