1.5 4 20 129 ? 10505

Options

A. 1044
B. 1048
C. 1548
D. 1052

Explanation

To solve the series 1.5, 4, 20, 129, ?, 10505, we need to identify the pattern in the sequence. Let's break it down step by step.

### Step 1: Identify the pattern
We start with the numbers: 1.5, 4, 20, 129, ?, 10505. The numbers are increasing significantly, suggesting a multiplication and addition pattern.

**Hint:** Look for a multiplication factor and an addition that changes with each step.

### Step 2: Analyze the first transition (1.5 to 4)
To go from 1.5 to 4:
- Multiply 1.5 by 2:
\[ 1.5 \times 2 = 3 \]
- Then add 1:
\[ 3 + 1 = 4 \]

**Hint:** Check if the multiplication factor increases with each step.

### Step 3: Analyze the second transition (4 to 20)
To go from 4 to 20:
- Multiply 4 by 3:
\[ 4 \times 3 = 12 \]
- Then add 8 (which is \(2^3\)):
\[ 12 + 8 = 20 \]

**Hint:** Notice the multiplication factor is increasing (2, 3, ...).

### Step 4: Analyze the third transition (20 to 129)
To go from 20 to 129:
- Multiply 20 by 6:
\[ 20 \times 6 = 120 \]
- Then add 9 (which is \(3^2\)):
\[ 120 + 9 = 129 \]

**Hint:** The multiplication factor is now 6. What could be the next multiplication factor?

### Step 5: Predict the next number (129 to ?)
Following the identified pattern:
- The next multiplication factor should be 8.
- Multiply 129 by 8:
\[ 129 \times 8 = 1032 \]
- Then add 16 (which is \(4^2\)):
\[ 1032 + 16 = 1048 \]

**Hint:** This gives us the next number in the series.

### Step 6: Verify the next transition (1048 to 10505)
To go from 1048 to 10505:
- Multiply 1048 by 10:
\[ 1048 \times 10 = 10480 \]
- Then add 25 (which is \(5^2\)):
\[ 10480 + 25 = 10505 \]

**Hint:** This confirms our pattern holds true.

### Conclusion
The missing number in the series is **1048**.

### Final Answer
The required number is **1048**.