Sum of the series `1+2.2+3.2^2 +4.2^3+.....+100.2^99` is

Options

A. `99.2^(100) - 1 `
B. `100.2^(100)`
C. `99.2^(100)`
D. `99.2^(100) + 1`

Answer

To find the sum of the series \( S = 1 + 2 \cdot 2 + 3 \cdot 2^2 + 4 \cdot 2^3 + \ldots + 100 \cdot 2^{99} \), we can follow these steps:

### Step 1: Define the series
Let:
\[ S = 1 + 2 \cdot 2 + 3 \cdot 2^2 + 4 \cdot 2^3 + \ldots + 100 \cdot 2^{99} \]

### Step 2: Multiply the series by 2
Now, multiply the entire series \( S \) by 2:
\[ 2S = 2 \cdot 1 + 2 \cdot 2 \cdot 2 + 3 \cdot 2^2 \cdot 2 + 4 \cdot 2^3 \cdot 2 + \ldots + 100 \cdot 2^{99} \cdot 2 \]
This simplifies to:
\[ 2S = 2 + 4 \cdot 2 + 6 \cdot 2^2 + 8 \cdot 2^3 + \ldots + 200 \cdot 2^{99} \]

### Step 3: Rearrange the series
Now we can express \( S \) and \( 2S \) together:
\[ S - 2S = 1 + (2 \cdot 2 - 2) + (3 \cdot 2^2 - 4 \cdot 2) + (4 \cdot 2^3 - 6 \cdot 2^2) + \ldots + (100 \cdot 2^{99} - 200 \cdot 2^{98}) \]
This gives us:
\[ -S = 1 + 2 + 2^2 + 2^3 + \ldots + 2^{99} - 100 \cdot 2^{100} \]

### Step 4: Identify the geometric series
The series \( 1 + 2 + 2^2 + 2^3 + \ldots + 2^{99} \) is a geometric series with:
- First term \( a = 1 \)
- Common ratio \( r = 2 \)
- Number of terms \( n = 100 \)

The sum of a geometric series is given by:
\[ \text{Sum} = \frac{a(r^n - 1)}{r - 1} \]
Substituting the values:
\[ \text{Sum} = \frac{1(2^{100} - 1)}{2 - 1} = 2^{100} - 1 \]

### Step 5: Substitute back into the equation
Now we can substitute this back into our equation:
\[ -S = (2^{100} - 1) - 100 \cdot 2^{100} \]
This simplifies to:
\[ -S = 2^{100} - 1 - 100 \cdot 2^{100} = -99 \cdot 2^{100} - 1 \]

### Step 6: Solve for S
Now, multiplying both sides by -1 gives:
\[ S = 99 \cdot 2^{100} + 1 \]

### Final Answer
Thus, the sum of the series is:
\[ S = 99 \cdot 2^{100} + 1 \]