3 4 ? 21 85 110 326

Options

A. 4
B. 10
C. 12
D. 14

Explanation

To find the missing number in the series: 3, 4, ?, 21, 85, 110, 326, we will analyze the pattern in the series step by step.

### Step 1: Identify the pattern
We will first look at the differences between consecutive numbers in the series.

- The first difference is:
\( 4 - 3 = 1 \)

- The second difference is:
\( ? - 4 \) (we will calculate this later)

- The third difference is:
\( 21 - ? \) (we will calculate this later)

- The fourth difference is:
\( 85 - 21 = 64 \)

- The fifth difference is:
\( 110 - 85 = 25 \)

- The sixth difference is:
\( 326 - 110 = 216 \)

### Step 2: Analyze the differences
Now we have the differences:
- From 3 to 4: \( 1 \)
- From 4 to ?: \( ? - 4 \)
- From ? to 21: \( 21 - ? \)
- From 21 to 85: \( 64 \)
- From 85 to 110: \( 25 \)
- From 110 to 326: \( 216 \)

### Step 3: Recognize the pattern in differences
The differences we have calculated seem to follow a pattern of squares and cubes:
- \( 1 = 1^2 \)
- \( 64 = 4^3 \)
- \( 25 = 5^2 \)
- \( 216 = 6^3 \)

This suggests that the pattern alternates between squares and cubes.

### Step 4: Fill in the missing differences
Following the pattern:
- The difference from 4 to the missing number should be \( 2^3 = 8 \)
- The difference from the missing number to 21 should be \( 3^2 = 9 \)

### Step 5: Calculate the missing number
Now we can calculate the missing number:
1. From 4 to the missing number:
\[ ? = 4 + 8 = 12 \]

2. From the missing number (12) to 21:
\[ 21 - 12 = 9 \]

### Conclusion
Thus, the missing number in the series is **12**.

### Final Answer
The missing number is **12**.

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