The coefficient of ` x^(101)` in the expansion of
` (1 - x) (1- 2x) (1 - 2^(2) x) - (1 - 2^(101) x)` is

Options

A. `2^(4950) - 2^(5050)`
B. `2^(5051) - 2^(5152)`
C. `2^(4950) - 2^(5051)`
D. `2^(5050) - 2^(5152)`

Answer

To find the coefficient of \( x^{101} \) in the expansion of

\[ (1 - x)(1 - 2x)(1 - 2^2 x) \cdots (1 - 2^{101} x) - (1 - 2^{101} x), \]

we can follow these steps:

### Step 1: Define the function
Let

\[ f(x) = (1 - x)(1 - 2x)(1 - 2^2 x) \cdots (1 - 2^{101} x). \]

### Step 2: Understand the maximum power of \( x \)
The maximum power of \( x \) in \( f(x) \) is \( x^{102} \) because there are 102 factors in the product (from \( 1 - 2^0 x \) to \( 1 - 2^{101} x \)).

### Step 3: Find the coefficient of \( x^{101} \)
The coefficient of \( x^{101} \) in \( f(x) \) can be found by considering the contributions from choosing 101 terms from the 102 factors. This can be computed as:

\[ \text{Coefficient of } x^{101} = -\sum_{i=0}^{101} 2^i, \]

where \( i \) is the index of the term we are excluding.

### Step 4: Calculate the sum
The sum \( \sum_{i=0}^{101} 2^i \) is a geometric series. The formula for the sum of a geometric series is:

\[ S_n = a \frac{(r^n - 1)}{(r - 1)}, \]

where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. Here, \( a = 1 \), \( r = 2 \), and \( n = 102 \):

\[ \sum_{i=0}^{101} 2^i = 1 \cdot \frac{(2^{102} - 1)}{(2 - 1)} = 2^{102} - 1. \]

### Step 5: Substitute back into the expression
Thus, the coefficient of \( x^{101} \) in \( f(x) \) is:

\[ -\sum_{i=0}^{101} 2^i = -(2^{102} - 1) = -2^{102} + 1. \]

### Step 6: Adjust for the subtraction of \( (1 - 2^{101} x) \)
Now we need to consider the subtraction of \( (1 - 2^{101} x) \). The coefficient of \( x^{101} \) in \( (1 - 2^{101} x) \) is \( -2^{101} \).

### Step 7: Combine the results
The final coefficient of \( x^{101} \) in the entire expression is:

\[ (-2^{102} + 1) - (-2^{101}) = -2^{102} + 1 + 2^{101}. \]

### Step 8: Simplify the expression
We can factor out \( -2^{101} \):

\[ -2^{101}(2 - 1) + 1 = -2^{101} + 1. \]

### Final Answer
Thus, the coefficient of \( x^{101} \) in the expansion is:

\[ -2^{101} + 1. \]