Arrange the expansion of `(x^(1//2) + (1)/(2x^(1//4)))` in decreasing powers of x. Suppose the coefficient of the first three terms form an arithemetic progression. Then the number of terms in the expression having integer powers of x is -

Options

A. 0
B. 2
C. 4
D. 8

Answer

To solve the problem, we need to expand the expression \( (x^{1/2} + \frac{1}{2} x^{-1/4})^n \) and analyze the coefficients of the first three terms to determine the number of terms with integer powers of \( x \).

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression:
\[ (x^{1/2} + \frac{1}{2} x^{-1/4})^n \]

2. **Use the Binomial Theorem**:
According to the Binomial Theorem, the expansion of \( (a + b)^n \) is given by:
\[ \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]
Here, \( a = x^{1/2} \) and \( b = \frac{1}{2} x^{-1/4} \).

3. **Write the General Term**:
The general term \( T_r \) in the expansion can be expressed as:
\[ T_r = \binom{n}{r} (x^{1/2})^{n-r} \left(\frac{1}{2} x^{-1/4}\right)^r \]
Simplifying this gives:
\[ T_r = \binom{n}{r} \frac{1}{2^r} x^{(n-r)/2 - r/4} \]
\[ = \binom{n}{r} \frac{1}{2^r} x^{(n - 2r)/4} \]

4. **Determine the Power of \( x \)**:
The power of \( x \) in the \( r \)-th term is:
\[ \frac{n - 2r}{4} \]

5. **Find Integer Powers of \( x \)**:
For \( \frac{n - 2r}{4} \) to be an integer, \( n - 2r \) must be divisible by 4. This leads to:
\[ n - 2r = 4k \quad \text{for some integer } k \]
Rearranging gives:
\[ 2r = n - 4k \quad \Rightarrow \quad r = \frac{n - 4k}{2} \]

6. **Determine Valid Values of \( r \)**:
Since \( r \) must be a non-negative integer, we need:
\[ n - 4k \geq 0 \quad \Rightarrow \quad k \leq \frac{n}{4} \]
Additionally, \( n - 4k \) must be even, which implies \( n \) and \( k \) must have the same parity (both even or both odd).

7. **Count the Possible Values of \( k \)**:
Let \( n = 4m + r \) where \( r \) is the remainder when \( n \) is divided by 4 (0, 1, 2, or 3). The possible values of \( k \) range from 0 to \( \lfloor n/4 \rfloor \).

8. **Final Count of Terms**:
The number of valid \( k \) values is:
- If \( n \) is even, \( k \) can take values \( 0, 1, 2, \ldots, \frac{n}{4} \) (total \( \frac{n}{4} + 1 \) values).
- If \( n \) is odd, \( k \) can take values \( 0, 1, 2, \ldots, \lfloor \frac{n}{4} \rfloor \) (total \( \lfloor \frac{n}{4} \rfloor + 1 \) values).

### Conclusion:
The number of terms in the expression having integer powers of \( x \) is given by the number of valid \( k \) values, which depends on whether \( n \) is even or odd.