Find the coefficient of `x^(7)` in the expansion of `(2x^(2)-(1)/(x))^(20)`

Explanation

To find the coefficient of \( x^7 \) in the expansion of \( (2x^2 - \frac{1}{x})^{20} \), we can follow these steps:

### Step 1: Identify the general term in the binomial expansion
The general term (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, \( a = 2x^2 \), \( b = -\frac{1}{x} \), and \( n = 20 \). Thus, the \( (r+1)^{th} \) term is:
\[ T_{r+1} = \binom{20}{r} (2x^2)^{20-r} \left(-\frac{1}{x}\right)^r \]

### Step 2: Simplify the general term
Now, we simplify \( T_{r+1} \):
\[ T_{r+1} = \binom{20}{r} (2^{20-r} (x^2)^{20-r}) \left(-\frac{1}{x}\right)^r \]
This can be rewritten as:
\[ T_{r+1} = \binom{20}{r} 2^{20-r} (-1)^r x^{2(20-r)} x^{-r} \]
Combining the powers of \( x \):
\[ T_{r+1} = \binom{20}{r} 2^{20-r} (-1)^r x^{40 - 2r - r} = \binom{20}{r} 2^{20-r} (-1)^r x^{40 - 3r} \]

### Step 3: Set the exponent of \( x \) to 7
We need the exponent of \( x \) to equal 7:
\[ 40 - 3r = 7 \]
Solving for \( r \):
\[ 40 - 7 = 3r \implies 33 = 3r \implies r = 11 \]

### Step 4: Substitute \( r \) back into the general term
Now, we substitute \( r = 11 \) into the general term:
\[ T_{12} = \binom{20}{11} 2^{20-11} (-1)^{11} x^{40 - 3 \cdot 11} \]
Calculating the components:
\[ T_{12} = \binom{20}{11} 2^9 (-1)^{11} x^{7} \]

### Step 5: Calculate the coefficient
The coefficient of \( x^7 \) is:
\[ \text{Coefficient} = \binom{20}{11} 2^9 (-1)^{11} \]
Since \( (-1)^{11} = -1 \):
\[ \text{Coefficient} = -\binom{20}{11} 2^9 \]

### Step 6: Calculate \( \binom{20}{11} \) and \( 2^9 \)
Using the binomial coefficient:
\[ \binom{20}{11} = \frac{20!}{11!(20-11)!} = \frac{20!}{11!9!} = 167960 \]
And calculating \( 2^9 \):
\[ 2^9 = 512 \]
Thus, the coefficient is:
\[ \text{Coefficient} = -167960 \times 512 \]

### Final Calculation
Calculating the final value:
\[ -167960 \times 512 = -85983220 \]

### Final Answer
The coefficient of \( x^7 \) in the expansion of \( (2x^2 - \frac{1}{x})^{20} \) is:
\[ \text{Coefficient} = -85983220 \]