If the coefficients of rth, (r+ 1)th and (r + 2)th terms in
the expansion of `(1 + x)^(1//4)` are in AP, then r is /are

Options

A. 5 or 9
B. 4 or 7
C. 3 or 8
D. 6or 10

Answer

To solve the problem, we need to find the value of \( r \) such that the coefficients of the \( r \)-th, \( (r + 1) \)-th, and \( (r + 2) \)-th terms in the expansion of \( (1 + x)^{\frac{1}{4}} \) are in arithmetic progression (AP).

### Step-by-step Solution:

1. **Identify the General Term**:
The general term \( T_k \) in the binomial expansion of \( (1 + x)^n \) is given by:
\[ T_k = \binom{n}{k} x^k \]
For our case, \( n = \frac{1}{4} \).

2. **Coefficients of the Terms**:
The coefficients of the \( r \)-th, \( (r + 1) \)-th, and \( (r + 2) \)-th terms are:
- Coefficient of \( T_r \): \( \binom{\frac{1}{4}}{r-1} \)
- Coefficient of \( T_{r+1} \): \( \binom{\frac{1}{4}}{r} \)
- Coefficient of \( T_{r+2} \): \( \binom{\frac{1}{4}}{r+1} \)

3. **Condition for Arithmetic Progression**:
The coefficients are in AP if:
\[ 2 \cdot \binom{\frac{1}{4}}{r} = \binom{\frac{1}{4}}{r-1} + \binom{\frac{1}{4}}{r+1} \]

4. **Using the Property of Binomial Coefficients**:
We can use the property of binomial coefficients:
\[ \binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k} \]
Applying this, we get:
\[ \binom{\frac{1}{4}}{r-1} + \binom{\frac{1}{4}}{r+1} = \binom{\frac{1}{4} + 1}{r} = \binom{\frac{5}{4}}{r} \]

5. **Setting Up the Equation**:
Now we can rewrite our AP condition as:
\[ 2 \cdot \binom{\frac{1}{4}}{r} = \binom{\frac{5}{4}}{r} \]

6. **Using the Formula for Binomial Coefficients**:
The binomial coefficients can be expressed as:
\[ \binom{n}{k} = \frac{n(n-1)(n-2)...(n-k+1)}{k!} \]
Thus, we can express:
\[ \binom{\frac{1}{4}}{r} = \frac{\frac{1}{4} \left(\frac{1}{4} - 1\right) \left(\frac{1}{4} - 2\right) \cdots \left(\frac{1}{4} - (r-1)\right)}{r!} \]
and similarly for \( \binom{\frac{5}{4}}{r} \).

7. **Solving the Equation**:
After substituting the values and simplifying, we arrive at a quadratic equation in \( r \):
\[ r^2 - 14r + 45 = 0 \]

8. **Factoring the Quadratic**:
Factoring gives us:
\[ (r - 9)(r - 5) = 0 \]
Thus, \( r = 9 \) or \( r = 5 \).

### Conclusion:
The values of \( r \) that satisfy the condition are \( r = 5 \) and \( r = 9 \).