Complete the following series 1, - 8, 81, ?, 15625

Options

A. `-1022`
B. `-1024`
C. `-4094`
D. `-4096`

Answer

To complete the series 1, -8, 81, ?, 15625, we need to identify the pattern in the sequence.

### Step-by-Step Solution:

1. **Identify the first term**: The first term is \(1\), which can be expressed as:
\[ 1 = (-1)^0 \cdot 1^1 \]

2. **Identify the second term**: The second term is \(-8\). This can be expressed as:
\[ -8 = (-1)^1 \cdot 2^3 \]

3. **Identify the third term**: The third term is \(81\). This can be expressed as:
\[ 81 = (-1)^0 \cdot 3^4 \]

4. **Identify the pattern**: The pattern seems to alternate between negative and positive values, and the bases of the powers increase by 1 each time:
- The first term is \(1 = (-1)^0 \cdot 1^1\)
- The second term is \(-8 = (-1)^1 \cdot 2^3\)
- The third term is \(81 = (-1)^0 \cdot 3^4\)

5. **Determine the fourth term**: Following the established pattern, the fourth term should be:
\[ (-1)^1 \cdot 4^5 \]
Now, calculate \(4^5\):
\[ 4^5 = 1024 \]
Therefore, the fourth term is:
\[ -1024 \]

6. **Identify the fifth term**: The fifth term is \(15625\). This can be expressed as:
\[ 15625 = (-1)^0 \cdot 5^6 \]
where \(5^6 = 15625\).

### Conclusion:
The complete series is:
\[ 1, -8, 81, -1024, 15625 \]