If the sequences `1,2,2,4,4,4,4,8,8,8,8,8,8,8,8…..` where n consecutive terms has value n then `1026^(th)`

Options

A. `2^(y)`
B. `2^(10)`
C. `2^(11)`
D. `2n+2`

Answer

To find the 1026th term in the sequence `1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8...`, where the value of `n` appears `n` times, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Pattern**:
- The sequence starts with 1 appearing once, 2 appearing twice, 3 appearing three times, and so on.
- The general term can be described as follows: The number `2^k` appears `2^k` times for `k = 0, 1, 2, ...`.

2. **Identify the Number of Terms**:
- We need to find how many terms are there up to a certain `n`.
- The total number of terms for the first `k` values can be calculated as:
\[ T_k = 1 + 2 + 4 + 8 + ... + 2^k = 2^{k+1} - 1 \]
- This formula represents the sum of a geometric series.

3. **Finding `k` such that `T_k < 1026`**:
- We need to find the largest `k` such that:
\[ 2^{k+1} - 1 < 1026 \]
- Solving for `k`:
\[ 2^{k+1} < 1027 \]
- Taking logarithm base 2:
\[ k + 1 < \log_2(1027) \]
- Approximating `log_2(1027)`:
\[ \log_2(1024) = 10 \quad \text{(since \(1024 = 2^{10}\))} \]
- Therefore, \(k + 1 < 10.01\) implies \(k \leq 9\).

4. **Calculate Total Terms for `k = 9`**:
- For \(k = 9\):
\[ T_9 = 2^{10} - 1 = 1023 \]
- This means the first 1023 terms consist of numbers from `1` to `2^9 = 512`.

5. **Finding the 1026th Term**:
- The next value after 1023 terms is \(2^{10} = 1024\), which appears `1024` times.
- The terms from 1024 to 2047 will all be `1024`.

6. **Conclusion**:
- The 1026th term falls within the range of terms that are all `1024`.
- Therefore, the value of the 1026th term is **1024**.

### Final Answer:
The value of the 1026th term in the sequence is **1024**.