Let n be an integer and `x,y,z lt1`. Suppose
`Delta=|(x^(n+1),x^(n+2),x^(n+3)),(y^(n+1),y^(n+2),y^(n+3)),(z^(n+1),z^(n+2),z^(n+3))|`
If `Delta=(x-y)(y-z)(z-x)x^(2)y^(2)z^(2)` then n is equal to

Options

A. `-1`
B. `0`
C. `1`
D. `2`

Answer

To solve the problem, we need to evaluate the determinant \( \Delta \) given by:

\[ \Delta = \begin{vmatrix} x^{n+1} & x^{n+2} & x^{n+3} \\ y^{n+1} & y^{n+2} & y^{n+3} \\ z^{n+1} & z^{n+2} & z^{n+3} \end{vmatrix} \]

We are also given that:

\[ \Delta = (x-y)(y-z)(z-x)x^2y^2z^2 \]

### Step 1: Calculate the Determinant

The determinant of a \( 3 \times 3 \) matrix can be calculated using the formula:

\[ \Delta = a(ei-fh) - b(di-fg) + c(dh-eg) \]

For our determinant, we can express it as follows:

\[ \Delta = x^{n+1} \begin{vmatrix} y^{n+2} & y^{n+3} \\ z^{n+2} & z^{n+3} \end{vmatrix} - x^{n+2} \begin{vmatrix} y^{n+1} & y^{n+3} \\ z^{n+1} & z^{n+3} \end{vmatrix} + x^{n+3} \begin{vmatrix} y^{n+1} & y^{n+2} \\ z^{n+1} & z^{n+2} \end{vmatrix} \]

Calculating the \( 2 \times 2 \) determinants:

1. \( \begin{vmatrix} y^{n+2} & y^{n+3} \\ z^{n+2} & z^{n+3} \end{vmatrix} = y^{n+2}z^{n+3} - y^{n+3}z^{n+2} = (y^{n+2} - y^{n+3})(z^{n+3} - z^{n+2}) \)

2. \( \begin{vmatrix} y^{n+1} & y^{n+3} \\ z^{n+1} & z^{n+3} \end{vmatrix} = y^{n+1}z^{n+3} - y^{n+3}z^{n+1} \)

3. \( \begin{vmatrix} y^{n+1} & y^{n+2} \\ z^{n+1} & z^{n+2} \end{vmatrix} = y^{n+1}z^{n+2} - y^{n+2}z^{n+1} \)

### Step 2: Combine the Results

After calculating the determinants, we can combine them into one expression for \( \Delta \):

\[ \Delta = x^{n+1} \cdot (y^{n+2} - y^{n+3})(z^{n+3} - z^{n+2}) - x^{n+2} \cdot (y^{n+1}z^{n+3} - y^{n+3}z^{n+1}) + x^{n+3} \cdot (y^{n+1}z^{n+2} - y^{n+2}z^{n+1}) \]

### Step 3: Compare with Given Expression

We know that:

\[ \Delta = (x-y)(y-z)(z-x)x^2y^2z^2 \]

### Step 4: Equate the Powers of \( x, y, z \)

From the expression of \( \Delta \), we can see that the highest power of \( x, y, z \) must match on both sides. The left-hand side will have terms involving \( x^{n+1}, y^{n+2}, z^{n+3} \) and so on.

### Step 5: Determine \( n \)

By comparing the coefficients and powers of \( x, y, z \) on both sides, we can find \( n \).

From the structure of the determinant, we can conclude that the highest power of \( x, y, z \) on the left-hand side must equal \( 2 \) from the right-hand side. Thus, we have:

\[ n + 2 = 2 \implies n = 0 \]

However, we also need to consider the polynomial degree and the structure of the factors \( (x-y)(y-z)(z-x) \) which leads us to conclude that \( n = 2 \).

### Final Answer

Thus, the value of \( n \) is:

\[ \boxed{2} \]