Find the middle term in the expansion of `(3x-(1)/(2x))^(16)`

Explanation

To find the middle term in the expansion of \((3x - \frac{1}{2x})^{16}\), we can follow these steps:

### Step 1: Identify the value of \(n\)
In the expression \((3x - \frac{1}{2x})^{16}\), the exponent \(n\) is 16.

### Step 2: Determine the middle term
Since \(n\) is even, the middle term can be found using the formula:
\[ \text{Middle Term} = \frac{n}{2} + 1 \]
Substituting \(n = 16\):
\[ \text{Middle Term} = \frac{16}{2} + 1 = 8 + 1 = 9 \]
Thus, we need to find the 9th term of the expansion.

### Step 3: Use the Binomial Theorem
According to the Binomial Theorem, the \(r + 1\)th term in the expansion of \((a + b)^n\) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, \(a = 3x\), \(b = -\frac{1}{2x}\), and \(n = 16\). For the 9th term, we have \(r = 8\).

### Step 4: Substitute values into the formula
Now we can write the 9th term:
\[ T_9 = \binom{16}{8} (3x)^{16-8} \left(-\frac{1}{2x}\right)^8 \]
This simplifies to:
\[ T_9 = \binom{16}{8} (3x)^8 \left(-\frac{1}{2}\right)^8 (x^{-8}) \]

### Step 5: Simplify the expression
Now we simplify:
\[ T_9 = \binom{16}{8} (3^8 x^8) \left(-\frac{1}{2^8}\right) x^{-8} \]
The \(x^8\) and \(x^{-8}\) cancel out:
\[ T_9 = \binom{16}{8} \cdot 3^8 \cdot \left(-\frac{1}{2^8}\right) \]

### Step 6: Calculate \(\binom{16}{8}\) and simplify further
Calculating \(\binom{16}{8}\):
\[ \binom{16}{8} = \frac{16!}{8!8!} = 12870 \]
Now substituting this value:
\[ T_9 = 12870 \cdot 3^8 \cdot \left(-\frac{1}{256}\right) \]
Calculating \(3^8\):
\[ 3^8 = 6561 \]
Thus:
\[ T_9 = 12870 \cdot 6561 \cdot \left(-\frac{1}{256}\right) \]

### Step 7: Final calculation
Calculating \(12870 \cdot 6561\):
\[ 12870 \cdot 6561 = 84459657 \]
Now divide by 256:
\[ T_9 = -\frac{84459657}{256} = -330000.25 \]

### Conclusion
The middle term in the expansion of \((3x - \frac{1}{2x})^{16}\) is:
\[ T_9 = -330000.25 \]