The 4th and 7th terms of a G.P. are 18 and 486 respectively. Find the G.P.

Explanation

To solve the problem, we need to find the first term (A) and the common ratio (R) of the geometric progression (G.P.) given that the 4th term is 18 and the 7th term is 486.

### Step-by-step Solution:

1. **Identify the Terms**:
The n-th term of a G.P. can be expressed as:
\[ T_n = A \cdot R^{n-1} \]
Therefore, the 4th term \( T_4 \) and the 7th term \( T_7 \) can be expressed as:
\[ T_4 = A \cdot R^{3} = 18 \quad (1) \]
\[ T_7 = A \cdot R^{6} = 486 \quad (2) \]

2. **Set Up the Equations**:
From the above equations, we have:
- Equation (1): \( A \cdot R^{3} = 18 \)
- Equation (2): \( A \cdot R^{6} = 486 \)

3. **Divide the Equations**:
To eliminate \( A \), we can divide equation (2) by equation (1):
\[ \frac{A \cdot R^{6}}{A \cdot R^{3}} = \frac{486}{18} \]
This simplifies to:
\[ R^{3} = \frac{486}{18} \]

4. **Calculate \( R^{3} \)**:
Now, calculate \( \frac{486}{18} \):
\[ R^{3} = 27 \]

5. **Find \( R \)**:
Taking the cube root of both sides gives:
\[ R = 3 \]

6. **Substitute \( R \) Back to Find \( A \)**:
Now, substitute \( R = 3 \) back into equation (1) to find \( A \):
\[ A \cdot (3^{3}) = 18 \]
\[ A \cdot 27 = 18 \]
\[ A = \frac{18}{27} = \frac{2}{3} \]

7. **Write the G.P.**:
Now that we have \( A \) and \( R \), we can write the terms of the G.P.:
- First term: \( A = \frac{2}{3} \)
- Second term: \( AR = \frac{2}{3} \cdot 3 = 2 \)
- Third term: \( AR^2 = \frac{2}{3} \cdot 3^2 = 6 \)

Therefore, the G.P. is:
\[ \frac{2}{3}, 2, 6, \ldots \]

### Final Answer:
The geometric progression is:
\[ \frac{2}{3}, 2, 6, \ldots \]