Let n be positive integer. If the coefficients of 2nd, 3rd and 4th terms in the expansion of `(1 + x)^(n)` are in A.P., then the value of n is

Options

A. 2
B. 5
C. 7
D. 9

Answer

To solve the problem, we need to find the positive integer \( n \) such that the coefficients of the 2nd, 3rd, and 4th terms in the expansion of \( (1 + x)^n \) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \( T_{r+1} \) in the expansion of \( (1 + x)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} x^r \]
Therefore, the coefficients of the terms can be expressed as:
- Coefficient of the 2nd term \( T_2 \): \( \binom{n}{1} \)
- Coefficient of the 3rd term \( T_3 \): \( \binom{n}{2} \)
- Coefficient of the 4th term \( T_4 \): \( \binom{n}{3} \)

2. **Set Up the A.P. Condition**:
For the coefficients to be in A.P., the condition is:
\[ 2 \cdot \binom{n}{2} = \binom{n}{1} + \binom{n}{3} \]

3. **Substitute the Binomial Coefficients**:
Substitute the values of the binomial coefficients:
- \( \binom{n}{1} = n \)
- \( \binom{n}{2} = \frac{n(n-1)}{2} \)
- \( \binom{n}{3} = \frac{n(n-1)(n-2)}{6} \)

The equation becomes:
\[ 2 \cdot \frac{n(n-1)}{2} = n + \frac{n(n-1)(n-2)}{6} \]
Simplifying this gives:
\[ n(n-1) = n + \frac{n(n-1)(n-2)}{6} \]

4. **Multiply Through by 6 to Eliminate the Denominator**:
Multiply the entire equation by 6:
\[ 6n(n-1) = 6n + n(n-1)(n-2) \]

5. **Rearrange the Equation**:
Rearranging gives:
\[ 6n^2 - 6n = 6n + n^3 - 3n^2 + 2n \]
Simplifying further:
\[ 6n^2 - 6n = n^3 - n^2 + 2n \]
\[ 0 = n^3 - 7n^2 + 8n \]

6. **Factor the Polynomial**:
Factor out \( n \):
\[ n(n^2 - 7n + 8) = 0 \]
This gives:
\[ n = 0 \quad \text{or} \quad n^2 - 7n + 8 = 0 \]

7. **Solve the Quadratic Equation**:
Solving \( n^2 - 7n + 8 = 0 \) using the quadratic formula:
\[ n = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} \]
\[ n = \frac{7 \pm \sqrt{49 - 32}}{2} \]
\[ n = \frac{7 \pm \sqrt{17}}{2} \]

The roots are:
\[ n = 7 \quad \text{and} \quad n = 1 \]

8. **Conclusion**:
Since \( n \) must be a positive integer, the possible values of \( n \) are \( n = 2 \) and \( n = 7 \).

### Final Answer:
The values of \( n \) are \( n = 2 \) and \( n = 7 \).