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

Explanation

To prove that \( x, y, z \) are in a geometric progression (G.P.) given that the 4th, 10th, and 16th terms of a G.P. are \( x, y, z \) respectively, we can follow these steps:

### Step 1: Define the terms of the G.P.
Let the first term of the G.P. be \( A \) and the common ratio be \( R \). The \( n \)-th term of a G.P. is given by the formula:
\[ A_n = A \cdot R^{n-1} \]

### Step 2: Write the expressions for the 4th, 10th, and 16th terms
Using the formula for the \( n \)-th term:
- The 4th term \( A_4 \) is:
\[ A_4 = A \cdot R^{4-1} = A \cdot R^3 = x \]
- The 10th term \( A_{10} \) is:
\[ A_{10} = A \cdot R^{10-1} = A \cdot R^9 = y \]
- The 16th term \( A_{16} \) is:
\[ A_{16} = A \cdot R^{16-1} = A \cdot R^{15} = z \]

### Step 3: Express \( A \) in terms of \( x \), \( y \), and \( z \)
From the equations derived:
1. From \( x = A \cdot R^3 \), we can express \( A \):
\[ A = \frac{x}{R^3} \]
2. From \( y = A \cdot R^9 \):
\[ A = \frac{y}{R^9} \]
3. From \( z = A \cdot R^{15} \):
\[ A = \frac{z}{R^{15}} \]

### Step 4: Set the expressions for \( A \) equal to each other
Since all three expressions equal \( A \), we can set them equal:
\[ \frac{x}{R^3} = \frac{y}{R^9} = \frac{z}{R^{15}} \]

### Step 5: Cross-multiply to find the relationship between \( x, y, z \)
From the first two equalities:
\[ \frac{x}{R^3} = \frac{y}{R^9} \implies x \cdot R^9 = y \cdot R^3 \implies y \cdot R^3 = x \cdot R^9 \]
From the second and third equalities:
\[ \frac{y}{R^9} = \frac{z}{R^{15}} \implies y \cdot R^{15} = z \cdot R^9 \implies z \cdot R^9 = y \cdot R^{15} \]

### Step 6: Prove that \( y^2 = xz \)
From the relationships derived:
1. \( y \cdot R^3 = x \cdot R^9 \)
2. \( z \cdot R^9 = y \cdot R^{15} \)

Now, we can express \( y^2 \):
\[ y^2 = (x \cdot R^9) \cdot R^6 = x \cdot R^{15} \]
And since \( z = A \cdot R^{15} \):
\[ y^2 = xz \]

### Conclusion
Since we have shown that \( y^2 = xz \), it follows that \( x, y, z \) are in G.P.