If `[.]` denotes the greatest integer function, then thevalue of natural number n satisfying the equation `[log_2 1]+[log_2 2]+[log_2 3]+.....+[log_2 n] = 1538`, is

Options

A. 255
B. 256
C. 254
D. 313

Explanation

To solve the equation

\[ [\log_2 1] + [\log_2 2] + [\log_2 3] + \ldots + [\log_2 n] = 1538, \]

we need to analyze the behavior of the greatest integer function applied to logarithms.

### Step-by-step Solution:

1. **Understanding the Greatest Integer Function**:
- The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).
- For logarithms, we have:
- \([\log_2 1] = 0\) (since \(\log_2 1 = 0\))
- \([\log_2 2] = 1\) (since \(\log_2 2 = 1\))
- \([\log_2 3] = 1\) (since \(\log_2 3\) is slightly less than 2)
- \([\log_2 4] = 2\) (since \(\log_2 4 = 2\))
- \([\log_2 5] = 2\) (since \(\log_2 5\) is slightly less than 3)
- \([\log_2 6] = 2\)
- \([\log_2 7] = 2\)
- \([\log_2 8] = 3\)
- Continuing this way, we see a pattern based on powers of 2.

2. **Identifying Ranges**:
- For \(k = 0\): \(1\) term contributes \(0\) (from \(1\)).
- For \(k = 1\): \(2\) terms contribute \(1\) (from \(2\) to \(3\)).
- For \(k = 2\): \(4\) terms contribute \(2\) (from \(4\) to \(7\)).
- For \(k = 3\): \(8\) terms contribute \(3\) (from \(8\) to \(15\)).
- For \(k = 4\): \(16\) terms contribute \(4\) (from \(16\) to \(31\)).
- For \(k = 5\): \(32\) terms contribute \(5\) (from \(32\) to \(63\)).
- For \(k = 6\): \(64\) terms contribute \(6\) (from \(64\) to \(127\)).
- For \(k = 7\): \(128\) terms contribute \(7\) (from \(128\) to \(255\)).

3. **Calculating Contributions**:
- The contributions can be summarized as follows:
- \(0 \cdot 1 = 0\)
- \(1 \cdot 2 = 2\)
- \(2 \cdot 4 = 8\)
- \(3 \cdot 8 = 24\)
- \(4 \cdot 16 = 64\)
- \(5 \cdot 32 = 160\)
- \(6 \cdot 64 = 384\)
- \(7 \cdot 128 = 896\)

4. **Summing Contributions**:
- Total contribution from \(1\) to \(255\):
\[ 0 + 2 + 8 + 24 + 64 + 160 + 384 + 896 = 1538. \]

5. **Conclusion**:
- The equation holds true for \(n = 255\). Therefore, the value of \(n\) satisfying the equation is:

\[ \boxed{255}. \]