Find the coefficient of `x^(10)` in the expansion of `(1+2x)/((1-2x)^(2))`.

Answer

To find the coefficient of \( x^{10} \) in the expansion of \( \frac{1 + 2x}{(1 - 2x)^2} \), we can break it down into steps.

### Step 1: Rewrite the expression
We start by rewriting the expression:
\[ \frac{1 + 2x}{(1 - 2x)^2} = (1 + 2x) \cdot (1 - 2x)^{-2} \]

### Step 2: Expand \( (1 - 2x)^{-2} \)
Using the binomial theorem, we can expand \( (1 - 2x)^{-2} \):
\[ (1 - 2x)^{-2} = \sum_{n=0}^{\infty} \binom{n + 1}{1} (2x)^n = \sum_{n=0}^{\infty} (n + 1)(2^n x^n) \]
This series gives us the coefficients for \( x^n \).

### Step 3: Find the coefficient of \( x^9 \) in \( (1 - 2x)^{-2} \)
Since we need to find the coefficient of \( x^{10} \) in the product \( (1 + 2x)(1 - 2x)^{-2} \), we first need the coefficient of \( x^9 \) in \( (1 - 2x)^{-2} \):
\[ \text{Coefficient of } x^9 = (9 + 1)(2^9) = 10 \cdot 512 = 5120 \]

### Step 4: Find the coefficient of \( x^8 \) in \( (1 - 2x)^{-2} \)
Next, we find the coefficient of \( x^8 \) in \( (1 - 2x)^{-2} \):
\[ \text{Coefficient of } x^8 = (8 + 1)(2^8) = 9 \cdot 256 = 2304 \]

### Step 5: Combine results
Now we can combine the results:
- The contribution from \( 1 \) in \( (1 + 2x) \) gives us the coefficient of \( x^{10} \) from \( (1 - 2x)^{-2} \), which is \( 0 \).
- The contribution from \( 2x \) gives us \( 2 \) times the coefficient of \( x^9 \):
\[ 2 \cdot 5120 = 10240 \]

### Step 6: Final coefficient
Thus, the total coefficient of \( x^{10} \) in the expansion of \( \frac{1 + 2x}{(1 - 2x)^2} \) is:
\[ 0 + 10240 = 10240 \]

### Conclusion
The coefficient of \( x^{10} \) in the expansion of \( \frac{1 + 2x}{(1 - 2x)^2} \) is \( 10240 \).

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