The sum of the first 10 terms of `(3)/(2)+(5)/(4)+(9)/(8)+(17)/(16)+cdots` is

Options

A. `10-2^(10)`
B. `9-2^(-10)`
C. `11-2^(-10)`
D. none of these

Answer

To find the sum of the first 10 terms of the series \(\frac{3}{2} + \frac{5}{4} + \frac{9}{8} + \frac{17}{16} + \cdots\), we first need to identify the general term of the series.

### Step 1: Identify the General Term
The given series can be analyzed as follows:
- The numerators are: 3, 5, 9, 17, ...
- The denominators are: 2, 4, 8, 16, ...

The denominators can be expressed as powers of 2:
- \(2^1, 2^2, 2^3, 2^4, \ldots\)

The numerators appear to follow a pattern. Let's denote the \(n\)-th term of the series as \(T_n\):
- \(T_n = \frac{a_n}{b_n}\)
Where \(a_n\) is the numerator and \(b_n = 2^n\) is the denominator.

### Step 2: Finding the Numerator Sequence
The numerators can be observed as follows:
- \(a_1 = 3\)
- \(a_2 = 5\)
- \(a_3 = 9\)
- \(a_4 = 17\)

We can see that:
- \(a_1 = 2^1 + 1\)
- \(a_2 = 2^2 + 1\)
- \(a_3 = 2^3 + 1\)
- \(a_4 = 2^4 + 1\)

Thus, we can conclude that:
\[ a_n = 2^n + 1 \]

### Step 3: General Term of the Series
Now, we can express the \(n\)-th term of the series:
\[ T_n = \frac{a_n}{b_n} = \frac{2^n + 1}{2^n} = 1 + \frac{1}{2^n} \]

### Step 4: Sum of the First 10 Terms
The sum of the first 10 terms \(S_{10}\) can be calculated as follows:
\[ S_{10} = T_1 + T_2 + T_3 + \ldots + T_{10} = \sum_{n=1}^{10} T_n \]
Substituting the expression for \(T_n\):
\[ S_{10} = \sum_{n=1}^{10} \left(1 + \frac{1}{2^n}\right) \]
This can be split into two separate sums:
\[ S_{10} = \sum_{n=1}^{10} 1 + \sum_{n=1}^{10} \frac{1}{2^n} \]
Calculating the first sum:
\[ \sum_{n=1}^{10} 1 = 10 \]

### Step 5: Calculate the Second Sum
The second sum is a geometric series:
\[ \sum_{n=1}^{10} \frac{1}{2^n} = \frac{\frac{1}{2}(1 - (\frac{1}{2})^{10})}{1 - \frac{1}{2}} = 1 - \frac{1}{2^{10}} = 1 - \frac{1}{1024} = \frac{1023}{1024} \]

### Step 6: Combine the Results
Now, we can combine the results of both sums:
\[ S_{10} = 10 + \frac{1023}{1024} = \frac{10240}{1024} + \frac{1023}{1024} = \frac{10240 + 1023}{1024} = \frac{11263}{1024} \]

### Final Answer
Thus, the sum of the first 10 terms of the series is:
\[ \boxed{\frac{11263}{1024}} \]