If the sum of the series `1+(3)/(2)+(5)/(4)+(7)/(8)+……+((2n-1))/((2)^(n-1))` is `f(n)`, then the value of `f(8)` is

Options

A. `4+(12)/(2^(5))`
B. `5+(13)/(2^(7))`
C. `6-(19)/(2^(7))`
D. `5-(13)/(2^(7))`

Answer

To solve the problem, we need to find the value of the function \( f(n) \) defined by the sum of the series

\[ f(n) = 1 + \frac{3}{2} + \frac{5}{4} + \frac{7}{8} + \ldots + \frac{2n-1}{2^{n-1}}. \]

### Step 1: Write the series in summation notation
The series can be expressed as:

\[ f(n) = \sum_{k=1}^{n} \frac{2k-1}{2^{k-1}}. \]

### Step 2: Multiply the series by \( \frac{1}{2} \)
To simplify the calculation, we will multiply the entire series by \( \frac{1}{2} \):

\[ \frac{1}{2} f(n) = \sum_{k=1}^{n} \frac{2k-1}{2^k}. \]

### Step 3: Shift the index of the series
Now, we can shift the index of the series in \( \frac{1}{2} f(n) \):

\[ \frac{1}{2} f(n) = \frac{1}{2} + \frac{3}{4} + \frac{5}{8} + \frac{7}{16} + \ldots + \frac{2n-1}{2^n}. \]

### Step 4: Subtract the two series
Now we subtract the shifted series from the original series:

\[ f(n) - \frac{1}{2} f(n) = \left(1 + \frac{3}{2} + \frac{5}{4} + \frac{7}{8} + \ldots + \frac{2n-1}{2^{n-1}}\right) - \left(\frac{1}{2} + \frac{3}{4} + \frac{5}{8} + \frac{7}{16} + \ldots + \frac{2n-1}{2^n}\right). \]

### Step 5: Simplify the equation
This gives us:

\[ \frac{1}{2} f(n) = 1 + \left(\frac{3}{2} - \frac{1}{2}\right) + \left(\frac{5}{4} - \frac{3}{4}\right) + \left(\frac{7}{8} - \frac{5}{8}\right) + \ldots + \left(\frac{2n-1}{2^{n-1}} - \frac{2n-1}{2^n}\right). \]

This simplifies to:

\[ \frac{1}{2} f(n) = 1 + 1 + 1 + \ldots + \frac{2n-1}{2^n}. \]

### Step 6: Recognize the geometric series
The remaining terms form a geometric series. The sum of the first \( n \) terms of a geometric series where the first term is \( \frac{1}{2} \) and the common ratio is \( \frac{1}{2} \) can be calculated using the formula:

\[ S_n = a \frac{1 - r^n}{1 - r} = \frac{1/2(1 - (1/2)^n)}{1/2} = 1 - \frac{1}{2^n}. \]

### Step 7: Substitute back to find \( f(n) \)
Thus, we have:

\[ \frac{1}{2} f(n) = n - \frac{1}{2^n}. \]

Multiplying through by 2 gives:

\[ f(n) = 2n - \frac{2}{2^n} = 2n - \frac{1}{2^{n-1}}. \]

### Step 8: Calculate \( f(8) \)
Now we can substitute \( n = 8 \):

\[ f(8) = 2(8) - \frac{1}{2^{7}} = 16 - \frac{1}{128} = 16 - 0.0078125 = 15.9921875. \]

### Final Answer
Thus, the value of \( f(8) \) is:

\[ \boxed{15.9921875}. \]