The fifth term of a G.P. is 81 and its second term is 24. Find the geometric progression.

Answer

To solve the problem, we need to find the geometric progression (G.P.) given that the fifth term is 81 and the second term is 24. Let's denote the first term of the G.P. as \( a \) and the common ratio as \( r \).

### Step-by-Step Solution:

1. **Identify the terms of the G.P.**:
- The terms of a G.P. can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Fifth term: \( ar^4 \)

2. **Set up the equations based on the given information**:
- From the problem, we know:
- \( ar^4 = 81 \) (Equation 1)
- \( ar = 24 \) (Equation 2)

3. **Divide Equation 1 by Equation 2**:
- We can eliminate \( a \) by dividing Equation 1 by Equation 2:
\[ \frac{ar^4}{ar} = \frac{81}{24} \]
- This simplifies to:
\[ r^3 = \frac{81}{24} \]

4. **Simplify the fraction**:
- Simplifying \( \frac{81}{24} \):
\[ \frac{81}{24} = \frac{27}{8} \]
- So we have:
\[ r^3 = \frac{27}{8} \]

5. **Take the cube root of both sides**:
- To find \( r \), take the cube root:
\[ r = \sqrt[3]{\frac{27}{8}} = \frac{3}{2} \]

6. **Substitute \( r \) back into Equation 2 to find \( a \)**:
- Now substitute \( r = \frac{3}{2} \) into Equation 2:
\[ a \left(\frac{3}{2}\right) = 24 \]
- Solving for \( a \):
\[ a = 24 \cdot \frac{2}{3} = 16 \]

7. **Write the G.P. using \( a \) and \( r \)**:
- Now that we have \( a = 16 \) and \( r = \frac{3}{2} \), we can write the terms of the G.P.:
- First term: \( a = 16 \)
- Second term: \( ar = 16 \cdot \frac{3}{2} = 24 \)
- Third term: \( ar^2 = 16 \cdot \left(\frac{3}{2}\right)^2 = 36 \)
- Fourth term: \( ar^3 = 16 \cdot \left(\frac{3}{2}\right)^3 = 54 \)
- Fifth term: \( ar^4 = 16 \cdot \left(\frac{3}{2}\right)^4 = 81 \)

8. **Final G.P.**:
- Therefore, the geometric progression is:
\[ 16, 24, 36, 54, 81 \]