If a term independent of x is to exist in the expansion of `(x + (1)/(x^2))^n` then n must be

Options

A. a multiple of 2
B. a multiple of 3
C. a multiple of 5
D. a multiple of 7

Answer

To find the value of \( n \) such that a term independent of \( x \) exists in the expansion of \( (x + \frac{1}{x^2})^n \), we can follow these steps:

### Step 1: Write the General Term
The general term in the expansion of \( (x + \frac{1}{x^2})^n \) can be expressed using the binomial theorem. The \( r \)-th term (denoted as \( T_r \)) is given by:

\[ T_r = \binom{n}{r} x^{n-r} \left(\frac{1}{x^2}\right)^r \]

### Step 2: Simplify the General Term
Now, we simplify \( T_r \):

\[ T_r = \binom{n}{r} x^{n-r} \cdot x^{-2r} = \binom{n}{r} x^{n - r - 2r} = \binom{n}{r} x^{n - 3r} \]

### Step 3: Set the Power of \( x \) to Zero
For the term to be independent of \( x \), the exponent of \( x \) must be zero:

\[ n - 3r = 0 \]

### Step 4: Solve for \( n \)
From the equation \( n - 3r = 0 \), we can solve for \( n \):

\[ n = 3r \]

### Step 5: Determine the Values of \( n \)
Since \( r \) can only be a non-negative integer (whole number), \( n \) must be a multiple of 3. Thus, \( n \) can take values like \( 0, 3, 6, 9, \ldots \).

### Final Answer
Therefore, \( n \) must be a multiple of 3.

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