In the sequence 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, ……, where n consecutive terms have the value n, the 1025th term is:

Options

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

Answer

To find the 1025th term in the sequence where n consecutive terms have the value n, we can follow these steps:

### Step 1: Understand the Sequence
The sequence is structured such that:
- The first term is 1 (1 time)
- The next two terms are 2 (2 times)
- The next four terms are 4 (4 times)
- The next eight terms are 8 (8 times)
- The next sixteen terms are 16 (16 times)
- And so on...

### Step 2: Identify the Pattern
The pattern shows that the value of the term is \(2^k\) where \(k\) is the number of times the value appears. The number of terms for each \(k\) is \(2^k\).

### Step 3: Calculate the Cumulative Count of Terms
Now, we calculate how many terms are there up to each \(k\):
- For \(k=0\): \(2^0 = 1\) term (Total = 1)
- For \(k=1\): \(2^1 = 2\) terms (Total = 1 + 2 = 3)
- For \(k=2\): \(2^2 = 4\) terms (Total = 3 + 4 = 7)
- For \(k=3\): \(2^3 = 8\) terms (Total = 7 + 8 = 15)
- For \(k=4\): \(2^4 = 16\) terms (Total = 15 + 16 = 31)
- For \(k=5\): \(2^5 = 32\) terms (Total = 31 + 32 = 63)
- For \(k=6\): \(2^6 = 64\) terms (Total = 63 + 64 = 127)
- For \(k=7\): \(2^7 = 128\) terms (Total = 127 + 128 = 255)
- For \(k=8\): \(2^8 = 256\) terms (Total = 255 + 256 = 511)
- For \(k=9\): \(2^9 = 512\) terms (Total = 511 + 512 = 1023)
- For \(k=10\): \(2^{10} = 1024\) terms (Total = 1023 + 1024 = 2047)

### Step 4: Determine Where the 1025th Term Falls
From the cumulative counts, we see:
- The 1023rd term is the last term of \(2^9\) (which is 512).
- The 1024th term is the first term of \(2^{10}\) (which is 1024).
- The 1025th term is the second term of \(2^{10}\) (which is also 1024).

### Conclusion
Thus, the 1025th term in the sequence is **1024**.

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