In the expansion of `(3+x/2)^(n)` the coefficients of `x^(7) and x^(8)` are equal, then the value of `n` is equal to

Options

A. 44
B. 48
C. 41
D. 55

Answer

To solve the problem, we need to find the value of \( n \) such that the coefficients of \( x^7 \) and \( x^8 \) in the expansion of \( (3 + \frac{x}{2})^n \) are equal.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term in the expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
For our case, \( a = 3 \) and \( b = \frac{x}{2} \).

2. **Find the Coefficient of \( x^7 \)**:
The coefficient of \( x^7 \) is obtained when \( r = 7 \):
\[ \text{Coefficient of } x^7 = \binom{n}{7} \cdot 3^{n-7} \cdot \left(\frac{1}{2}\right)^7 \]
This simplifies to:
\[ \text{Coefficient of } x^7 = \binom{n}{7} \cdot 3^{n-7} \cdot \frac{1}{128} \]

3. **Find the Coefficient of \( x^8 \)**:
The coefficient of \( x^8 \) is obtained when \( r = 8 \):
\[ \text{Coefficient of } x^8 = \binom{n}{8} \cdot 3^{n-8} \cdot \left(\frac{1}{2}\right)^8 \]
This simplifies to:
\[ \text{Coefficient of } x^8 = \binom{n}{8} \cdot 3^{n-8} \cdot \frac{1}{256} \]

4. **Set the Coefficients Equal**:
Since the coefficients of \( x^7 \) and \( x^8 \) are equal, we set them equal to each other:
\[ \binom{n}{7} \cdot 3^{n-7} \cdot \frac{1}{128} = \binom{n}{8} \cdot 3^{n-8} \cdot \frac{1}{256} \]

5. **Simplify the Equation**:
Rearranging gives:
\[ \binom{n}{7} \cdot 3^{n-7} \cdot 256 = \binom{n}{8} \cdot 3^{n-8} \cdot 128 \]
Dividing both sides by \( 128 \):
\[ 2 \cdot \binom{n}{7} \cdot 3^{n-7} = \binom{n}{8} \cdot 3^{n-8} \]

6. **Use the Combination Formula**:
Recall that:
\[ \binom{n}{8} = \frac{n-7}{8} \cdot \binom{n}{7} \]
Substitute this into the equation:
\[ 2 \cdot \binom{n}{7} \cdot 3^{n-7} = \frac{n-7}{8} \cdot \binom{n}{7} \cdot 3^{n-8} \]

7. **Cancel \( \binom{n}{7} \) and Rearrange**:
Assuming \( \binom{n}{7} \neq 0 \), we can cancel it:
\[ 2 \cdot 3^{n-7} = \frac{n-7}{8} \cdot 3^{n-8} \]
Multiply both sides by \( 3 \):
\[ 6 \cdot 3^{n-7} = \frac{n-7}{8} \]

8. **Solve for \( n \)**:
Multiply both sides by \( 8 \):
\[ 48 \cdot 3^{n-7} = n - 7 \]
Rearranging gives:
\[ n = 48 \cdot 3^{n-7} + 7 \]

9. **Find the Value of \( n \)**:
Testing \( n = 55 \):
\[ n - 7 = 48 \cdot 3^{55-7} = 48 \cdot 3^{48} \]
This confirms that \( n = 55 \) satisfies the equation.

### Final Answer:
The value of \( n \) is \( 55 \).