The total number of terms which depand on the value of
x in the expansion of ` (x^(2) - 2+ (1)/(x^(2)))^(n)` is

Options

A. ` 2n+1`
B. 2n
C. ` n+ 1`
D. n

Explanation

To find the total number of terms that depend on the value of \( x \) in the expansion of \( (x^2 - 2 + \frac{1}{x^2})^n \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ (x^2 - 2 + \frac{1}{x^2})^n \]

### Step 2: Identify the Terms
In the expression, we have three terms: \( x^2 \), \( -2 \), and \( \frac{1}{x^2} \).

### Step 3: Binomial Expansion
Using the multinomial expansion, the general term in the expansion can be represented as:
\[ \frac{n!}{a!b!c!} (x^2)^a (-2)^b \left(\frac{1}{x^2}\right)^c \]
where \( a + b + c = n \).

### Step 4: Simplify the General Term
The general term simplifies to:
\[ \frac{n!}{a!b!c!} (-2)^b x^{2a - 2c} \]
Here, \( 2a - 2c \) represents the power of \( x \).

### Step 5: Determine the Power of \( x \)
The power of \( x \) can be expressed as:
\[ x^{2a - 2c} \]
To find the total number of distinct terms that depend on \( x \), we need to determine the values that \( 2a - 2c \) can take.

### Step 6: Find the Range of \( 2a - 2c \)
Since \( a + b + c = n \), we can express \( c \) as \( c = n - a - b \). Thus:
\[ 2a - 2c = 2a - 2(n - a - b) = 2a - 2n + 2a + 2b = 4a + 2b - 2n \]
This means the power of \( x \) can take values depending on the values of \( a \) and \( b \).

### Step 7: Determine the Minimum and Maximum Values
- The minimum value occurs when \( a = 0 \) and \( b = n \), giving:
\[ 2(0) - 2(n) = -2n \]
- The maximum value occurs when \( a = n \) and \( b = 0 \), giving:
\[ 2(n) - 2(0) = 2n \]

### Step 8: Calculate the Number of Distinct Terms
The distinct powers of \( x \) range from \( -2n \) to \( 2n \). The total number of distinct integer values from \( -2n \) to \( 2n \) is:
\[ 2n - (-2n) + 1 = 4n + 1 \]

### Final Answer
Thus, the total number of terms that depend on the value of \( x \) in the expansion is:
\[ \boxed{4n + 1} \]