The coeffiicent of `x^(n)` in the binomial expansion of `( 1-x)^(-2)` is

Options

A. `(2^(n))/(2!)`
B. (n+1)
C. n
D. 2n

Answer

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

### Step 1: Understand the Binomial Expansion
The binomial expansion for \( (1 - x)^{-n} \) is given by:
\[ (1 - x)^{-n} = \sum_{r=0}^{\infty} \binom{n + r - 1}{r} x^r \]
where \( \binom{n + r - 1}{r} \) is the binomial coefficient.

### Step 2: Substitute \( n = 2 \)
In our case, we need to find the coefficient of \( x^n \) in \( (1 - x)^{-2} \). Here, we substitute \( n = 2 \):
\[ (1 - x)^{-2} = \sum_{r=0}^{\infty} \binom{2 + r - 1}{r} x^r = \sum_{r=0}^{\infty} \binom{r + 1}{r} x^r \]

### Step 3: Simplify the Binomial Coefficient
The binomial coefficient \( \binom{r + 1}{r} \) can be simplified:
\[ \binom{r + 1}{r} = \frac{(r + 1)!}{r! \cdot 1!} = r + 1 \]

### Step 4: Write the Series
Now we can write the series:
\[ (1 - x)^{-2} = \sum_{r=0}^{\infty} (r + 1) x^r \]

### Step 5: Find the Coefficient of \( x^n \)
The coefficient of \( x^n \) in this expansion is simply \( n + 1 \). This is because when \( r = n \), the term is \( (n + 1) x^n \).

### Conclusion
Thus, the coefficient of \( x^n \) in the expansion of \( (1 - x)^{-2} \) is:
\[ \boxed{n + 1} \]

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