The Sum `sum_(K=1)^(20) K1/(2^(K))` is equal to.

Options

A. `2-3/2^(17)`
B. `1-(11)/2^(20)`
C. `2-(11)/(2^(19)`
D. `2-(21)/2^(20)`

Answer

To solve the summation \( S = \sum_{k=1}^{20} \frac{k}{2^k} \), we can follow these steps:

### Step 1: Write the summation explicitly
We start with the expression for \( S \):
\[ S = \frac{1}{2^1} + \frac{2}{2^2} + \frac{3}{2^3} + \frac{4}{2^4} + \ldots + \frac{20}{2^{20}} \]

### Step 2: Multiply the summation by \( \frac{1}{2} \)
Now, we multiply the entire summation \( S \) by \( \frac{1}{2} \):
\[ \frac{S}{2} = \frac{1}{2^2} + \frac{2}{2^3} + \frac{3}{2^4} + \ldots + \frac{20}{2^{21}} \]

### Step 3: Subtract the two equations
Next, we subtract \( \frac{S}{2} \) from \( S \):
\[ S - \frac{S}{2} = S \left(1 - \frac{1}{2}\right) = \frac{S}{2} \]
This gives us:
\[ S - \frac{S}{2} = \left(\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \ldots + \frac{1}{2^{20}}\right) - \left(\frac{1}{2^2} + \frac{2}{2^3} + \frac{3}{2^4} + \ldots + \frac{20}{2^{21}}\right) \]

### Step 4: Simplify the left side
The left side simplifies to:
\[ \frac{S}{2} = \frac{1}{2} + \left(\frac{2}{2^2} - \frac{1}{2^2}\right) + \left(\frac{3}{2^3} - \frac{2}{2^3}\right) + \ldots + \left(\frac{20}{2^{20}} - \frac{19}{2^{20}}\right) - \frac{20}{2^{21}} \]
This results in:
\[ \frac{S}{2} = \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \ldots + \frac{1}{2^{20}} - \frac{20}{2^{21}} \]

### Step 5: Calculate the sum of the geometric series
The series \( \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \ldots + \frac{1}{2^{20}} \) is a geometric series with first term \( a = \frac{1}{2} \) and common ratio \( r = \frac{1}{2} \). The sum of the first \( n \) terms of a geometric series is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
For our series:
\[ S = \frac{1/2 \left(1 - (1/2)^{20}\right)}{1 - 1/2} = 1 - \frac{1}{2^{20}} \]

### Step 6: Substitute back to find \( S \)
Now we substitute this back into our equation:
\[ \frac{S}{2} = \left(1 - \frac{1}{2^{20}}\right) - \frac{20}{2^{21}} \]
This simplifies to:
\[ \frac{S}{2} = 1 - \frac{1 + 20}{2^{20}} = 1 - \frac{21}{2^{20}} \]

### Step 7: Solve for \( S \)
Multiply both sides by 2:
\[ S = 2 - \frac{42}{2^{20}} = 2 - \frac{21}{2^{19}} \]

### Final Answer
Thus, the final value of the summation is:
\[ S = 2 - \frac{21}{2^{19}} \]