Write the middle term or terms in the expansion of
(i) `(x^(2) - (1)/(x) )^6`

Answer

To find the middle term(s) in the expansion of \((x^2 - \frac{1}{x})^6\), we will follow these steps:

### Step 1: Identify the values of \(n\)
In the expression \((x^2 - \frac{1}{x})^6\), we have \(n = 6\).

**Hint:** The exponent in the binomial expression is key to determining the number of terms in the expansion.

### Step 2: Determine the number of terms
The number of terms in the expansion of \((x + y)^n\) is \(n + 1\). Therefore, for \(n = 6\), the number of terms is \(6 + 1 = 7\).

**Hint:** Remember that for any binomial expansion \((a + b)^n\), the number of terms is \(n + 1\).

### Step 3: Identify the middle term
Since \(n = 6\) is even, the middle term will be the \(\frac{n}{2} + 1\)th term, which is the \(4\)th term.

**Hint:** For even \(n\), the middle term is found using the formula \(\frac{n}{2} + 1\).

### Step 4: Write the general term
The general term \(T_{r+1}\) in the expansion of \((x^2 - \frac{1}{x})^n\) is given by:
\[ T_{r+1} = \binom{n}{r} (x^2)^{n-r} \left(-\frac{1}{x}\right)^r \]
For our case:
\[ T_{r+1} = \binom{6}{r} (x^2)^{6-r} \left(-\frac{1}{x}\right)^r \]

**Hint:** The general term formula is crucial for finding specific terms in the expansion.

### Step 5: Substitute \(r = 3\) for the 4th term
To find the 4th term, substitute \(r = 3\):
\[ T_{4} = \binom{6}{3} (x^2)^{6-3} \left(-\frac{1}{x}\right)^3 \]
Calculating this gives:
\[ T_{4} = \binom{6}{3} (x^2)^{3} \left(-\frac{1}{x}\right)^3 = \binom{6}{3} x^6 \cdot \left(-\frac{1}{x^3}\right) \]

**Hint:** Make sure to carefully apply the binomial coefficient and simplify the powers of \(x\).

### Step 6: Simplify the expression
Now, simplify:
\[ T_{4} = \binom{6}{3} x^{6 - 3} \cdot (-1) = \binom{6}{3} x^{3} \cdot (-1) \]
Calculating \(\binom{6}{3}\):
\[ \binom{6}{3} = \frac{6!}{3!3!} = 20 \]
Thus, we have:
\[ T_{4} = 20 \cdot (-1) \cdot x^{3} = -20x^{3} \]

### Conclusion
The middle term in the expansion of \((x^2 - \frac{1}{x})^6\) is:
\[ \boxed{-20x^{3}} \]