Find the middle term in the expanssion of (3x-x^3/6)^7

We are asked to find the middle term in the expansion of \((3x - \frac{x^3}{6})^7\).

### Step 1: Identify the binomial expansion formula
The binomial expansion of \((a + b)^n\) is given by:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

Here, we identify \(a = 3x\), \(b = -\frac{x^3}{6}\), and \(n = 7\).

### Step 2: Find the middle term index
In a binomial expansion of degree \(n\), the middle term corresponds to:

- If \(n\) is even, the middle term is the \((n/2 + 1)\)-th term.
- If \(n\) is odd, the middle term is the \(((n+1)/2)\)-th term.

Since \(n = 7\) is odd, the middle term is the \(((7+1)/2) = 4\)-th term.

### Step 3: Apply the binomial expansion formula for the 4th term
The \(k\)-th term in the binomial expansion is:

\[
T_{k+1} = \binom{n}{k} a^{n-k} b^k
\]

For the middle term, \(k = 3\):

\[
T_4 = \binom{7}{3} (3x)^{7-3} \left(-\frac{x^3}{6}\right)^3
\]

### Step 4: Calculate each part

- \(\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35\)
- \((3x)^4 = 3^4 x^4 = 81x^4\)
- \(\left(-\frac{x^3}{6}\right)^3 = \left(-\frac{1}{6}\right)^3 x^9 = -\frac{1}{216} x^9\)

### Step 5: Multiply everything together

\[
T_4 = 35 \times 81x^4 \times \left(-\frac{1}{216} x^9\right)
\]

Simplify:

\[
T_4 = 35 \times 81 \times -\frac{1}{216} x^{4+9}
\]
\[
T_4 = 35 \times -\frac{81}{216} x^{13}
\]
\[
T_4 = 35 \times -\frac{3}{8} x^{13}
\]
\[
T_4 = -\frac{105}{8} x^{13}
\]

### Final Answer:
The middle term in the expansion of \((3x - \frac{x^3}{6})^7\) is:

\[
-\frac{105}{8} x^{13}
\]

Would you like more details on any of the steps or clarification?

### Relative Questions:
1. How does the binomial expansion change for negative powers?
2. Can the binomial theorem be used for non-integer exponents?
3. What are the uses of binomial expansion in calculus?
4. How do you derive the general term in a binomial expansion?
5. What is the importance of combinations in the binomial theorem?

**Tip:** When solving binomial expansion problems, always check whether the exponent is odd or even to quickly find the middle term.

Find the middle term in the expansion of (3x - x^3/6)^7

Learn how to find the middle term in the binomial expansion of (3x - x^3/6)^7 with a step-by-step solution. This problem covers advanced algebraic concepts including binomial expansion, combinations, and exponentiation. Suitable for students in Grades 11-12.

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