The second term of a G.P. is 18 and the fifth term is 486. Find:
(i) the first term, (ii) the common ratio

Answer

To solve the problem, we need to find the first term (a) and the common ratio (r) of the given geometric progression (G.P.) where the second term is 18 and the fifth term is 486.

### Step-by-Step Solution:

1. **Understand the terms of a G.P.**:
- The n-th term of a G.P. can be expressed as:
\[ T_n = ar^{n-1} \]
- Here, \( a \) is the first term and \( r \) is the common ratio.

2. **Set up the equations**:
- Given that the second term \( T_2 = ar = 18 \) (1)
- Given that the fifth term \( T_5 = ar^4 = 486 \) (2)

3. **Express the equations**:
- From equation (1):
\[ ar = 18 \]
- From equation (2):
\[ ar^4 = 486 \]

4. **Divide the equations**:
- To eliminate \( a \), divide equation (2) by equation (1):
\[ \frac{ar^4}{ar} = \frac{486}{18} \]
- Simplifying this gives:
\[ r^3 = \frac{486}{18} \]

5. **Calculate \( r^3 \)**:
- Calculate \( \frac{486}{18} \):
\[ r^3 = 27 \]

6. **Find \( r \)**:
- Taking the cube root of both sides:
\[ r = 3 \]

7. **Substitute \( r \) back to find \( a \)**:
- Substitute \( r = 3 \) into equation (1):
\[ a \cdot 3 = 18 \]
- Solving for \( a \):
\[ a = \frac{18}{3} = 6 \]

### Final Answers:
- The first term \( a = 6 \)
- The common ratio \( r = 3 \)