If the coefficient of `x^(7)` and `x^(8)` in `(2+(x)/(3))^(n)` are equal, then n is

Options

A. 54
B. 55
C. 56
D. 58

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 expression \( \left(2 + \frac{x}{3}\right)^n \) are equal.

### Step-by-Step Solution:

1. **Identify the Coefficients**:
The general term in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
Here, \( a = 2 \) and \( b = \frac{x}{3} \).

2. **Coefficient of \( x^7 \)**:
For \( x^7 \), we have \( r = 7 \):
\[ \text{Coefficient of } x^7 = \binom{n}{7} \cdot 2^{n-7} \cdot \left(\frac{1}{3}\right)^7 \]
Thus,
\[ C_7 = \binom{n}{7} \cdot 2^{n-7} \cdot \frac{1}{2187} \]

3. **Coefficient of \( x^8 \)**:
For \( x^8 \), we have \( r = 8 \):
\[ \text{Coefficient of } x^8 = \binom{n}{8} \cdot 2^{n-8} \cdot \left(\frac{1}{3}\right)^8 \]
Thus,
\[ C_8 = \binom{n}{8} \cdot 2^{n-8} \cdot \frac{1}{6561} \]

4. **Set the Coefficients Equal**:
Since the coefficients are equal:
\[ \binom{n}{7} \cdot 2^{n-7} \cdot \frac{1}{2187} = \binom{n}{8} \cdot 2^{n-8} \cdot \frac{1}{6561} \]

5. **Simplify the Equation**:
Cancel out common terms:
\[ \binom{n}{7} \cdot 2^{n-7} \cdot 6561 = \binom{n}{8} \cdot 2^{n-8} \cdot 2187 \]
Rearranging gives:
\[ \binom{n}{7} \cdot 6561 = \binom{n}{8} \cdot 2 \cdot 2187 \]

6. **Use the Relationship Between Binomial Coefficients**:
We know that:
\[ \binom{n}{8} = \frac{n-7}{8} \cdot \binom{n}{7} \]
Substituting this into the equation:
\[ \binom{n}{7} \cdot 6561 = \frac{n-7}{8} \cdot \binom{n}{7} \cdot 2 \cdot 2187 \]
Cancel \( \binom{n}{7} \) (assuming \( n \geq 8 \)):
\[ 6561 = \frac{(n-7) \cdot 2 \cdot 2187}{8} \]

7. **Solve for \( n \)**:
Multiply both sides by 8:
\[ 8 \cdot 6561 = (n-7) \cdot 2 \cdot 2187 \]
Simplifying gives:
\[ 52488 = (n-7) \cdot 4374 \]
Dividing both sides by 4374:
\[ n - 7 = \frac{52488}{4374} = 12 \]
Thus,
\[ n = 12 + 7 = 19 \]

### Final Answer:
The value of \( n \) is \( 19 \).