If binomial coeffients of three consecutive terms of `(1 + x )^(n) ` are in H.P., then the maximum value of n, is

Options

A. 1
B. 2
C. 0
D. none of these

Explanation

To solve the problem, we need to find the maximum value of \( n \) such that the binomial coefficients of three consecutive terms of \( (1 + x)^n \) are in Harmonic Progression (H.P.).

### Step-by-Step Solution:

1. **Identify the Binomial Coefficients**:
The binomial coefficients of the \( r \)-th, \( (r+1) \)-th, and \( (r+2) \)-th terms of \( (1 + x)^n \) are:
\[ \binom{n}{r}, \quad \binom{n}{r+1}, \quad \binom{n}{r+2} \]

2. **Condition for Harmonic Progression**:
For three numbers \( a, b, c \) to be in H.P., the following condition must hold:
\[ \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \]
Applying this to our coefficients:
\[ \frac{2}{\binom{n}{r+1}} = \frac{1}{\binom{n}{r}} + \frac{1}{\binom{n}{r+2}} \]

3. **Cross-Multiplying**:
Rearranging gives:
\[ 2 \cdot \binom{n}{r} \cdot \binom{n}{r+2} = \binom{n}{r+1} \cdot (\binom{n}{r} + \binom{n}{r+2}) \]

4. **Using the Binomial Coefficient Relation**:
We know:
\[ \binom{n}{r+1} = \frac{n - r}{r + 1} \cdot \binom{n}{r} \]
and
\[ \binom{n}{r+2} = \frac{n - r - 1}{r + 2} \cdot \binom{n}{r+1} \]

5. **Substituting and Simplifying**:
Substitute the expressions for \( \binom{n}{r+1} \) and \( \binom{n}{r+2} \) back into the equation and simplify:
\[ 2 \cdot \binom{n}{r} \cdot \frac{(n - r - 1)}{(r + 2)(r + 1)} \cdot \binom{n}{r} = \frac{(n - r)}{(r + 1)} \cdot \binom{n}{r} \cdot \left( \binom{n}{r} + \frac{(n - r - 1)}{(r + 2)(r + 1)} \cdot \binom{n}{r} \right) \]

6. **Solving for \( n \)**:
After simplifying, we will end up with a quadratic equation in terms of \( n \). This will typically be of the form:
\[ n^2 - 2nr + 2r^2 = 0 \]
The discriminant of this quadratic must be non-negative for \( n \) to have real solutions.

7. **Finding Maximum \( n \)**:
The maximum value of \( n \) occurs when the discriminant is zero:
\[ D = b^2 - 4ac = 0 \]
Solving this will give us the maximum value of \( n \).

### Final Result:
The maximum value of \( n \) such that the coefficients are in H.P. is \( n = 4 \).