If the sequence `1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, ...`where `n` consecutive terms has value n then `1025^th` term is

Options

A. `2^(9)`
B. `2^(10)`
C. `2^(11)`
D. `2^(8)`

Explanation

To find the 1025th term of the sequence `1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, ...`, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Sequence**:
The sequence is defined such that the number `n` appears `n` times. For example:
- `1` appears `1` time
- `2` appears `2` times
- `4` appears `4` times
- `8` appears `8` times
- and so on...

2. **Identify the Pattern**:
The numbers in the sequence are powers of `2`. Specifically, the `n`-th term corresponds to `2^k` where `k` is the number of times `2^k` appears in the sequence.

3. **Calculate the Total Number of Terms**:
To find out how many terms are there up to a certain `k`, we can sum the series:
\[ 1 + 2 + 4 + 8 + ... + 2^k = 2^{k+1} - 1 \]
This is the sum of a geometric series.

4. **Set Up the Inequality**:
We need to find the largest `k` such that:
\[ 2^{k+1} - 1 < 1025 \]
This means:
\[ 2^{k+1} < 1026 \]

5. **Solve for `k`**:
Let's find the largest `k` satisfying the inequality:
- Calculate powers of `2`:
- \(2^{10} = 1024\)
- \(2^{11} = 2048\)
- Therefore, \(k + 1 = 10\) gives \(k = 9\) since \(2^{10} < 1026\).

6. **Find the Number of Terms**:
Now calculate the total number of terms up to `k = 9`:
\[ 2^{9+1} - 1 = 2^{10} - 1 = 1024 - 1 = 1023 \]
This means the first 1023 terms consist of:
- `1` (1 time)
- `2` (2 times)
- `4` (4 times)
- `8` (8 times)
- `16` (16 times)
- `32` (32 times)
- `64` (64 times)
- `128` (128 times)
- `256` (256 times)
- `512` (512 times)

7. **Determine the 1025th Term**:
Since the 1024th term is the last `512`, the 1025th term will be the first occurrence of `1024`, which is `2^{10}`.

### Final Answer:
Thus, the 1025th term of the sequence is:
\[ \boxed{1024} \]