3 4 16 75 364 ?

Options

A. 783
B. 828
C. 1293
D. 1945

Explanation

To find the missing number in the series: 3, 4, 16, 75, 364, ?, we will analyze the pattern step by step.

### Step 1: Identify the pattern in the series
We start with the numbers: 3, 4, 16, 75, 364. We need to determine how each number relates to the next.

### Step 2: Calculate the differences and ratios
- From 3 to 4: \(4 = 3 + 1\)
- From 4 to 16: \(16 = 4 \times 4\)
- From 16 to 75: \(75 = 16 \times 4 + 11\)
- From 75 to 364: \(364 = 75 \times 4 + 64\)

### Step 3: Look for a multiplication pattern
It appears that after the first addition, the series starts multiplying by 4 and then adding a certain number:
- \(3 + 1 = 4\)
- \(4 \times 4 = 16\)
- \(16 \times 4 + 11 = 75\)
- \(75 \times 4 + 64 = 364\)

### Step 4: Determine the next number
Following the pattern:
- The next number should be calculated as follows:
\[ 364 \times 4 + 125 \]
Here, 125 is \(5^3\) (the next cube in the sequence).

### Step 5: Perform the calculation
Calculating \(364 \times 4\):
\[ 364 \times 4 = 1456 \]
Now, add 125:
\[ 1456 + 125 = 1581 \]

### Conclusion
The next number in the series is **1581**.

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