Consider the binomial expansion of `(sqrt(x)+(1/(2x^(1/4))))^n n in NN`, where the terms of the expansion are written in decreasing powers of x. If the coefficients of the first three terms form an arithmetic progression then the statement(s) which hold good is(are) (A) total number of terms in the expansion of the binomial is 8 (B) number of terms in the expansion with integral power of x is 3 (C) there is no term in the expansion which is independent of x (D) fourth and fifth are the middle terms of the expansion

Options

A. Total number of terms in the expansion of the binomial is 8
B. Number of terms in the expansion with integral power of x is 3
C. There is no term in the expansion which in indepandent of x
D. Fourth and fifth are the middle terms of the expansion

Explanation

To solve the given problem, we will analyze the binomial expansion of \((\sqrt{x} + \frac{1}{2x^{1/4}})^n\) and determine the conditions under which the coefficients of the first three terms form an arithmetic progression. We will then evaluate the provided statements based on our findings.

### Step 1: Identify the Binomial Expansion

The binomial expansion of \((a + b)^n\) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, we have:
- \(a = \sqrt{x} = x^{1/2}\)
- \(b = \frac{1}{2} x^{-1/4}\)

Thus, the \(T_{r+1}\) term becomes:
\[ T_{r+1} = \binom{n}{r} \left(x^{1/2}\right)^{n-r} \left(\frac{1}{2} x^{-1/4}\right)^r \]
This simplifies to:
\[ T_{r+1} = \binom{n}{r} \frac{1}{2^r} x^{\frac{n}{2} - \frac{r}{4}} \]

### Step 2: Find the First Three Terms

Now we will find the first three terms \(T_1\), \(T_2\), and \(T_3\) by substituting \(r = 0\), \(1\), and \(2\) respectively.

1. **For \(r = 0\)**:
\[ T_1 = \binom{n}{0} \frac{1}{2^0} x^{\frac{n}{2}} = x^{\frac{n}{2}} \]

2. **For \(r = 1\)**:
\[ T_2 = \binom{n}{1} \frac{1}{2^1} x^{\frac{n}{2} - \frac{1}{4}} = n \cdot \frac{1}{2} x^{\frac{n}{2} - \frac{1}{4}} \]

3. **For \(r = 2\)**:
\[ T_3 = \binom{n}{2} \frac{1}{2^2} x^{\frac{n}{2} - \frac{2}{4}} = \frac{n(n-1)}{2} \cdot \frac{1}{4} x^{\frac{n}{2} - \frac{1}{2}} = \frac{n(n-1)}{8} x^{\frac{n}{2} - \frac{1}{2}} \]

### Step 3: Coefficients of the First Three Terms

The coefficients of the first three terms are:
- \(c_1 = 1\)
- \(c_2 = \frac{n}{2}\)
- \(c_3 = \frac{n(n-1)}{8}\)

### Step 4: Arithmetic Progression Condition

The coefficients \(c_1\), \(c_2\), and \(c_3\) form an arithmetic progression if:
\[ 2c_2 = c_1 + c_3 \]
Substituting the coefficients:
\[ 2 \cdot \frac{n}{2} = 1 + \frac{n(n-1)}{8} \]
This simplifies to:
\[ n = 1 + \frac{n(n-1)}{8} \]
Multiplying through by 8:
\[ 8n = 8 + n(n-1) \]
Rearranging gives:
\[ n^2 - 9n + 8 = 0 \]
Factoring:
\[ (n - 1)(n - 8) = 0 \]
Thus, \(n = 1\) or \(n = 8\). Since \(n\) must be a natural number greater than 1 for the expansion to have meaningful terms, we take \(n = 8\).

### Step 5: Analyze the Statements

1. **(A)** Total number of terms in the expansion of the binomial is \(n + 1 = 9\) (not 8). **False**
2. **(B)** Number of terms in the expansion with integral powers of \(x\): The powers of \(x\) are \(\frac{8}{2} - \frac{r}{4}\). For \(r = 0, 1, 2, \ldots, 8\), we find that \(r = 0, 4, 8\) yield integral powers, hence there are 3 integral terms. **True**
3. **(C)** Independent term means \(4 - \frac{3r}{2} = 0\), solving gives \(r = \frac{8}{3}\), which is not an integer. **True**
4. **(D)** The middle term for \(n = 8\) is the 5th term (since \(n + 1 = 9\)), not the 4th and 5th. **False**

### Conclusion

The correct statements are **(B)** and **(C)**.