The sum of series `(2^19 + 1/2^19) + 2(2^18 + 1/2^18 ) +3 (2^17 + 1/2^17) ` + …. + (19) `(2+1/2)` + 20 is :

Options

A. `2^21 - 4 + 1/2^19`
B. `2^21-16+1/2^18`
C. `2^21-24-1/2^19`
D. None of these

Answer

To solve the given series \( S = (2^{19} + \frac{1}{2^{19}}) + 2(2^{18} + \frac{1}{2^{18}}) + 3(2^{17} + \frac{1}{2^{17}}) + \ldots + 19(2 + \frac{1}{2}) + 20 \), we can break it down into two parts: the sum of the terms involving \( 2^n \) and the sum of the terms involving \( \frac{1}{2^n} \).

### Step 1: Define the Series
Let:
- \( x = 2^{19} + 2 \cdot 2^{18} + 3 \cdot 2^{17} + \ldots + 19 \cdot 2 + 20 \)
- \( y = \frac{1}{2^{19}} + \frac{2}{2^{18}} + \frac{3}{2^{17}} + \ldots + \frac{19}{2} \)

Thus, the total sum can be expressed as:
\[ S = x + y + 20 \]

### Step 2: Calculate \( x \)
To calculate \( x \), we can express it as:
\[ x = \sum_{k=1}^{20} k \cdot 2^{20-k} \]

Now, we can use the formula for the sum of the first \( n \) terms of a geometric series. We can also use the technique of multiplying the series by \( \frac{1}{2} \) to help simplify it.

1. Multiply \( x \) by \( \frac{1}{2} \):
\[ \frac{x}{2} = 2^{18} + 2 \cdot 2^{17} + 3 \cdot 2^{16} + \ldots + 19 \cdot 1 \]

2. Subtract this from \( x \):
\[ x - \frac{x}{2} = 2^{19} + (2-1) \cdot 2^{18} + (3-2) \cdot 2^{17} + \ldots + (19-18) \cdot 2 + 20 \]
\[ \frac{x}{2} = 2^{19} + 2^{18} + 2^{17} + \ldots + 2 + 20 \]

3. The left-hand side simplifies to:
\[ \frac{x}{2} = 2^{19} + 2^{18} + 2^{17} + \ldots + 2 + 20 \]

4. The sum of the geometric series \( 2^{19} + 2^{18} + \ldots + 2 \) can be calculated as:
\[ \text{Sum} = 2(1 - (1/2)^{19})/(1 - 1/2) = 2(1 - \frac{1}{2^{19}}) = 2 - \frac{1}{2^{18}} \]

5. Therefore, we can find \( x \):
\[ x = 2(2 - \frac{1}{2^{18}}) + 40 = 2^{21} - 42 \]

### Step 3: Calculate \( y \)
Now, calculate \( y \):
1. \( y = \sum_{k=1}^{19} \frac{k}{2^k} \)

Using a similar approach:
1. Multiply \( y \) by \( 2 \):
\[ 2y = \frac{1}{2^{18}} + \frac{2}{2^{17}} + \frac{3}{2^{16}} + \ldots + \frac{19}{2} \]

2. Subtract \( y \) from \( 2y \):
\[ y = \frac{1}{2^{19}} + \left(\frac{2}{2^{18}} - \frac{1}{2^{19}}\right) + \left(\frac{3}{2^{17}} - \frac{2}{2^{18}}\right) + \ldots + \left(\frac{19}{2} - \frac{18}{2}\right) \]

3. This gives:
\[ y = 1 - \frac{1}{2^{19}} - 19 \]

4. Therefore:
\[ y = 2 - 19 - \frac{1}{2^{19}} = -17 + \frac{1}{2^{19}} \]

### Step 4: Combine \( x \), \( y \), and \( 20 \)
Now substitute \( x \) and \( y \) back into \( S \):
\[ S = (2^{21} - 42) + (-17 + \frac{1}{2^{19}}) + 20 \]
\[ S = 2^{21} - 39 + \frac{1}{2^{19}} \]

### Final Result
Thus, the sum of the series is:
\[ S = 2^{21} - 39 + \frac{1}{2^{19}} \]