If 5, x, y, z, 405 are the first five terms of a geometric progression, find the values of x, y, and z.

Answer

To solve the problem, we need to find the values of \( x \), \( y \), and \( z \) in the geometric progression (GP) given the first five terms: \( 5, x, y, z, 405 \).

### Step-by-step Solution:

1. **Identify the first term and the common ratio**:
- The first term \( a \) of the GP is \( 5 \).
- The fifth term \( a r^4 \) is given as \( 405 \).

2. **Set up the equation for the fifth term**:
\[ a r^4 = 405 \]
Substituting \( a = 5 \):
\[ 5 r^4 = 405 \]

3. **Solve for \( r^4 \)**:
- Divide both sides by \( 5 \):
\[ r^4 = \frac{405}{5} = 81 \]

4. **Find \( r \)**:
- Since \( 81 = 3^4 \), we can conclude:
\[ r = 3 \]

5. **Calculate \( x \)**:
- The second term \( x \) is given by \( a r \):
\[ x = 5 \cdot 3 = 15 \]

6. **Calculate \( y \)**:
- The third term \( y \) is given by \( a r^2 \):
\[ y = 5 \cdot 3^2 = 5 \cdot 9 = 45 \]

7. **Calculate \( z \)**:
- The fourth term \( z \) is given by \( a r^3 \):
\[ z = 5 \cdot 3^3 = 5 \cdot 27 = 135 \]

### Final Values:
- \( x = 15 \)
- \( y = 45 \)
- \( z = 135 \)

### Summary:
The values of \( x \), \( y \), and \( z \) in the geometric progression are:
- \( x = 15 \)
- \( y = 45 \)
- \( z = 135 \)