The coefficient of `x^49` in the expansion of `(x-1)(x-1/2)(x-1/2^2)........(x-1/2^49)` is equal to

Options

A. `-2(1- 1/(2)^50 )`
B. `+ "ve coefficient of" x`
C. `-"ve coefficient of" x `
D. `-2(1-1/(2)^(49))`

Answer

To find the coefficient of \( x^{49} \) in the expansion of

\[ (x - 1)(x - \frac{1}{2})(x - \frac{1}{2^2}) \cdots (x - \frac{1}{2^{49}}), \]

we can follow these steps:

### Step 1: Identify the terms in the product
The product consists of 50 factors:
\[ (x - 1), (x - \frac{1}{2}), (x - \frac{1}{2^2}), \ldots, (x - \frac{1}{2^{49}}). \]

### Step 2: Understand the coefficient of \( x^{49} \)
The coefficient of \( x^{49} \) in the expansion is obtained by taking one constant term from one of the factors and \( x \) from the remaining 49 factors.

### Step 3: Write down the constant terms
The constant terms from each factor are:
- From \( (x - 1) \): \(-1\)
- From \( (x - \frac{1}{2}) \): \(-\frac{1}{2}\)
- From \( (x - \frac{1}{2^2}) \): \(-\frac{1}{4}\)
- From \( (x - \frac{1}{2^3}) \): \(-\frac{1}{8}\)
- ...
- From \( (x - \frac{1}{2^{49}}) \): \(-\frac{1}{2^{49}}\)

### Step 4: Sum the constant terms
The coefficient of \( x^{49} \) will be the sum of these constant terms taken with a negative sign:
\[ -1 - \frac{1}{2} - \frac{1}{4} - \frac{1}{8} - \ldots - \frac{1}{2^{49}}. \]

### Step 5: Factor out the negative sign
We can factor out \(-1\):
\[ -(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots + \frac{1}{2^{49}}). \]

### Step 6: Recognize the series as a geometric series
The series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots + \frac{1}{2^{49}} \) is a geometric series where:
- The first term \( a = 1 \)
- The common ratio \( r = \frac{1}{2} \)
- The number of terms \( n = 50 \)

### Step 7: Use the formula for the sum of a geometric series
The sum \( S_n \) of the first \( n \) terms of a geometric series is given by:
\[ S_n = a \frac{1 - r^n}{1 - r}. \]
Substituting the values:
\[ S_{50} = 1 \cdot \frac{1 - (\frac{1}{2})^{50}}{1 - \frac{1}{2}} = \frac{1 - \frac{1}{2^{50}}}{\frac{1}{2}} = 2(1 - \frac{1}{2^{50}}) = 2 - \frac{2}{2^{50}}. \]

### Step 8: Substitute back into the expression for the coefficient
Now substituting back, we have:
\[ \text{Coefficient of } x^{49} = -\left(2 - \frac{2}{2^{50}}\right) = -2 + \frac{2}{2^{50}}. \]

### Final Step: Simplify the expression
Thus, the coefficient of \( x^{49} \) in the expansion is:
\[ \frac{2}{2^{50}} - 2 = \frac{2 - 2^{51}}{2^{50}} = \frac{-2^{51} + 2}{2^{50}} = \frac{2(1 - 2^{50})}{2^{50}}. \]

### Conclusion
The coefficient of \( x^{49} \) in the expansion is:
\[ \frac{2(1 - 2^{50})}{2^{50}}. \]