If the coefficient of `x^11` and `x^12` in `(2+ (8x)/(3))^n` are equal find n

Answer

To solve the problem, we need to find the value of \( n \) such that the coefficients of \( x^{11} \) and \( x^{12} \) in the expression \( \left( 2 + \frac{8x}{3} \right)^n \) are equal.

### Step-by-step Solution:

1. **Identify the General Term**:
The general term in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_k = \binom{n}{k} a^{n-k} b^k \]
Here, \( a = 2 \) and \( b = \frac{8x}{3} \).

2. **Find the Coefficient of \( x^{11} \)**:
For \( x^{11} \), we set \( k = 11 \):
\[ T_{11} = \binom{n}{11} \cdot 2^{n-11} \cdot \left(\frac{8x}{3}\right)^{11} \]
The coefficient of \( x^{11} \) is:
\[ \text{Coefficient of } x^{11} = \binom{n}{11} \cdot 2^{n-11} \cdot \frac{8^{11}}{3^{11}} \]

3. **Find the Coefficient of \( x^{12} \)**:
For \( x^{12} \), we set \( k = 12 \):
\[ T_{12} = \binom{n}{12} \cdot 2^{n-12} \cdot \left(\frac{8x}{3}\right)^{12} \]
The coefficient of \( x^{12} \) is:
\[ \text{Coefficient of } x^{12} = \binom{n}{12} \cdot 2^{n-12} \cdot \frac{8^{12}}{3^{12}} \]

4. **Set the Coefficients Equal**:
Since the coefficients of \( x^{11} \) and \( x^{12} \) are equal, we have:
\[ \binom{n}{11} \cdot 2^{n-11} \cdot \frac{8^{11}}{3^{11}} = \binom{n}{12} \cdot 2^{n-12} \cdot \frac{8^{12}}{3^{12}} \]

5. **Simplify the Equation**:
Cancel \( \frac{8^{11}}{3^{11}} \) from both sides:
\[ \binom{n}{11} \cdot 2^{n-11} = \binom{n}{12} \cdot 2^{n-12} \cdot \frac{8}{3} \]
Rearranging gives:
\[ \binom{n}{11} \cdot 2^{n-11} = \binom{n}{12} \cdot 2^{n-12} \cdot \frac{8}{3} \]
Dividing both sides by \( 2^{n-12} \):
\[ \binom{n}{11} \cdot 2 = \binom{n}{12} \cdot \frac{8}{3} \]

6. **Use the Relationship Between Binomial Coefficients**:
Recall that:
\[ \binom{n}{12} = \frac{n-11}{12} \cdot \binom{n}{11} \]
Substitute this into the equation:
\[ \binom{n}{11} \cdot 2 = \frac{n-11}{12} \cdot \binom{n}{11} \cdot \frac{8}{3} \]

7. **Cancel \( \binom{n}{11} \)** (assuming \( n \geq 11 \)):
\[ 2 = \frac{(n-11) \cdot 8}{36} \]
Simplifying gives:
\[ 2 = \frac{2(n-11)}{9} \]
Multiply both sides by 9:
\[ 18 = 2(n-11) \]
Divide by 2:
\[ 9 = n - 11 \]
Thus:
\[ n = 20 \]

### Final Answer:
The value of \( n \) is \( 20 \).