If the tenth term of the sequence `S=1+5+13+29+…… ` is k, then `(k)/(500)` is equal to

Answer

To solve the problem, we need to find the 10th term of the sequence given by \( S = 1 + 5 + 13 + 29 + \ldots \) and then compute \( \frac{k}{500} \).

### Step-by-Step Solution:

1. **Identify the Sequence**:
The sequence starts with the terms:
- \( T_1 = 1 \)
- \( T_2 = 5 \)
- \( T_3 = 13 \)
- \( T_4 = 29 \)

2. **Find the Differences**:
Let's calculate the differences between consecutive terms:
- \( T_2 - T_1 = 5 - 1 = 4 \)
- \( T_3 - T_2 = 13 - 5 = 8 \)
- \( T_4 - T_3 = 29 - 13 = 16 \)

The differences are \( 4, 8, 16 \), which are \( 2^2, 2^3, 2^4 \). This suggests that the differences follow a pattern related to powers of 2.

3. **General Formula for the nth Term**:
Observing the differences, we can hypothesize that the nth term can be expressed as:
\[ T_n = 2^{n+1} - 3 \]
This is derived from the pattern of differences.

4. **Calculate the 10th Term**:
Now, we can find the 10th term using our formula:
\[ T_{10} = 2^{10+1} - 3 = 2^{11} - 3 \]
Calculate \( 2^{11} \):
\[ 2^{11} = 2048 \]
Therefore,
\[ T_{10} = 2048 - 3 = 2045 \]

5. **Compute \( \frac{k}{500} \)**:
Now, we need to find \( \frac{2045}{500} \):
\[ \frac{2045}{500} = 4.09 \]

### Final Answer:
Thus, \( \frac{k}{500} = 4.09 \).