Find the middle term (s) in the expansion of `((3x)/(7) - 2y)^10`

Answer

To find the middle term(s) in the expansion of \(\left(\frac{3x}{7} - 2y\right)^{10}\), we can follow these steps:

### Step 1: Identify the total number of terms
In the expansion of \((a + b)^n\), the total number of terms is \(n + 1\). Here, \(n = 10\), so the total number of terms is:
\[ 10 + 1 = 11 \]

### Step 2: Determine the middle term
Since the total number of terms is odd (11), the middle term will be the \(\frac{n + 1}{2}\)th term. Thus, the middle term is the \(6\)th term.

### Step 3: Write the general term
The general term \(T_r\) in the expansion of \((a + b)^n\) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
For our expression, \(a = \frac{3x}{7}\), \(b = -2y\), and \(n = 10\).

### Step 4: Substitute values into the general term
The \(r\)th term (where \(r = 5\) for the 6th term) is:
\[ T_6 = \binom{10}{5} \left(\frac{3x}{7}\right)^{10-5} \left(-2y\right)^5 \]

### Step 5: Calculate the binomial coefficient
First, we calculate \(\binom{10}{5}\):
\[ \binom{10}{5} = \frac{10!}{5!5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \]

### Step 6: Calculate the powers
Now, we calculate the powers:
\[ \left(\frac{3x}{7}\right)^5 = \frac{(3x)^5}{7^5} = \frac{243x^5}{16807} \]
\[ (-2y)^5 = -32y^5 \]

### Step 7: Combine all parts to find the middle term
Now we can substitute these values back into the expression for \(T_6\):
\[ T_6 = 252 \cdot \frac{243x^5}{16807} \cdot (-32y^5) \]
\[ = 252 \cdot \frac{243 \cdot (-32)x^5y^5}{16807} \]
\[ = \frac{-80640x^5y^5}{16807} \]

### Final Result
Thus, the middle term in the expansion of \(\left(\frac{3x}{7} - 2y\right)^{10}\) is:
\[ T_6 = \frac{-80640x^5y^5}{16807} \]

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