Let 2048 arithmetic means be inserted between ` 2^(22) - 1 and 2^(22) + 1` . Suppose the sum of these arithmetic
means is S . Then `(S)/(2^(23)) ` is ….

Answer

To solve the problem, we need to find the value of \( \frac{S}{2^{23}} \), where \( S \) is the sum of 2048 arithmetic means inserted between \( 2^{22} - 1 \) and \( 2^{22} + 1 \).

### Step-by-Step Solution:

1. **Identify the Terms**:
- The first term \( a = 2^{22} - 1 \)
- The last term \( l = 2^{22} + 1 \)
- The number of arithmetic means \( n = 2048 \)

2. **Total Number of Terms**:
- The total number of terms in the series, including the two endpoints and the arithmetic means, is \( n + 2 = 2048 + 2 = 2050 \).

3. **Finding the Common Difference \( d \)**:
- The formula for the last term in an arithmetic series is given by:
\[ l = a + (n - 1)d \]
- Substituting the known values:
\[ 2^{22} + 1 = (2^{22} - 1) + 2049d \]
- Simplifying this:
\[ 2^{22} + 1 - 2^{22} + 1 = 2049d \]
\[ 2 = 2049d \]
- Therefore, the common difference \( d \) is:
\[ d = \frac{2}{2049} \]

4. **Calculating the Sum of the Series**:
- The sum \( S_1 \) of the entire series can be calculated using the formula:
\[ S_n = \frac{n}{2} (a + l) \]
- Substituting the values:
\[ S_1 = \frac{2050}{2} \left( (2^{22} - 1) + (2^{22} + 1) \right) \]
\[ S_1 = 1025 \left( 2^{22} - 1 + 2^{22} + 1 \right) \]
\[ S_1 = 1025 \left( 2 \cdot 2^{22} \right) = 1025 \cdot 2^{23} \]

5. **Finding the Sum of the Arithmetic Means \( S \)**:
- The sum of the arithmetic means \( S \) is given by:
\[ S = S_1 - a - l \]
- Substituting the values:
\[ S = 1025 \cdot 2^{23} - (2^{22} - 1) - (2^{22} + 1) \]
- Simplifying:
\[ S = 1025 \cdot 2^{23} - 2^{22} + 1 - 2^{22} - 1 \]
\[ S = 1025 \cdot 2^{23} - 2 \cdot 2^{22} \]
\[ S = 1025 \cdot 2^{23} - 2^{23} = (1025 - 2) \cdot 2^{23} = 1024 \cdot 2^{23} \]

6. **Calculating \( \frac{S}{2^{23}} \)**:
- Now we can find \( \frac{S}{2^{23}} \):
\[ \frac{S}{2^{23}} = \frac{1024 \cdot 2^{23}}{2^{23}} = 1024 \]

### Final Answer:
\[ \frac{S}{2^{23}} = 1024 \]