If the 4th, 10th and 16th terms of a G.P. are x,y and z respectively, then x,y,z are in

Options

A. G.P.
B. A.P.
C. H.P.
D. A.G.P.

Answer

To solve the problem, we need to determine the relationship between the 4th, 10th, and 16th terms of a geometric progression (G.P.) given that they are represented by \( x \), \( y \), and \( z \) respectively.

### Step-by-Step Solution:

1. **Understand the nth term of a G.P.**:
The nth term of a geometric progression can be expressed as:
\[ T_n = a r^{n-1} \]
where \( a \) is the first term and \( r \) is the common ratio.

2. **Express the given terms**:
- The 4th term \( x \) can be expressed as:
\[ x = a r^{4-1} = a r^3 \]
- The 10th term \( y \) can be expressed as:
\[ y = a r^{10-1} = a r^9 \]
- The 16th term \( z \) can be expressed as:
\[ z = a r^{16-1} = a r^{15} \]

3. **Set up the ratios**:
To find the relationship between \( x \), \( y \), and \( z \), we can take the ratio of consecutive terms:
\[ \frac{y}{x} = \frac{a r^9}{a r^3} = r^{6} \]
and
\[ \frac{z}{y} = \frac{a r^{15}}{a r^9} = r^{6} \]

4. **Conclude the relationship**:
Since both ratios \( \frac{y}{x} \) and \( \frac{z}{y} \) are equal to \( r^6 \), we can conclude that:
\[ \frac{y}{x} = \frac{z}{y} \]
This implies that \( x, y, z \) are in a geometric progression.

5. **Final Answer**:
Therefore, \( x, y, z \) are in G.P. (Geometric Progression).