The second term of a GP is 24 and its fifth term is 81. Find the sum of its first five terms.

Explanation

To solve the problem, we need to find the sum of the first five terms of a Geometric Progression (GP) given that the second term is 24 and the fifth term is 81.

### Step-by-Step Solution:

1. **Identify the terms of the GP**:
- Let the first term be \( A \) and the common ratio be \( R \).
- The second term of the GP can be expressed as:
\[ AR = 24 \quad \text{(1)} \]
- The fifth term of the GP can be expressed as:
\[ AR^4 = 81 \quad \text{(2)} \]

2. **Divide the equations**:
- We can divide equation (2) by equation (1):
\[ \frac{AR^4}{AR} = \frac{81}{24} \]
- This simplifies to:
\[ R^3 = \frac{81}{24} \]
- Simplifying \( \frac{81}{24} \):
\[ R^3 = \frac{27}{8} \]

3. **Find the value of \( R \)**:
- Taking the cube root of both sides:
\[ R = \sqrt[3]{\frac{27}{8}} = \frac{3}{2} \]

4. **Substitute \( R \) back to find \( A \)**:
- Substitute \( R \) back into equation (1):
\[ A \cdot \frac{3}{2} = 24 \]
- Solving for \( A \):
\[ A = 24 \cdot \frac{2}{3} = 16 \]

5. **Calculate the sum of the first five terms**:
- The formula for the sum of the first \( n \) terms of a GP is:
\[ S_n = A \frac{R^n - 1}{R - 1} \]
- For the first five terms (\( n = 5 \)):
\[ S_5 = 16 \frac{(\frac{3}{2})^5 - 1}{\frac{3}{2} - 1} \]

6. **Calculate \( (\frac{3}{2})^5 \)**:
- Calculate \( (\frac{3}{2})^5 \):
\[ (\frac{3}{2})^5 = \frac{243}{32} \]

7. **Substitute back into the sum formula**:
- Now substitute this value back into the sum formula:
\[ S_5 = 16 \frac{\frac{243}{32} - 1}{\frac{1}{2}} \]
- Simplifying \( \frac{243}{32} - 1 \):
\[ \frac{243}{32} - 1 = \frac{243 - 32}{32} = \frac{211}{32} \]

8. **Final calculation**:
- Substitute this into the sum formula:
\[ S_5 = 16 \cdot \frac{\frac{211}{32}}{\frac{1}{2}} = 16 \cdot \frac{211}{32} \cdot 2 = 16 \cdot \frac{211}{16} = 211 \]

### Conclusion:
The sum of the first five terms of the GP is \( 211 \).