6 19 58 ? 214 331

Options

A. 113
B. 123
C. 133
D. 143

Explanation

To solve the series: 6, 19, 58, ?, 214, 331, we will find the missing number step by step.

### Step 1: Find the differences between consecutive terms.
- First, calculate the difference between the first two terms:
- \(19 - 6 = 13\)

- Next, calculate the difference between the second and third terms:
- \(58 - 19 = 39\)

- Now, calculate the difference between the last two terms:
- \(331 - 214 = 117\)

### Step 2: List the differences.
- The differences we have so far are:
- From 6 to 19: 13
- From 19 to 58: 39
- From 214 to 331: 117

### Step 3: Identify a pattern in the differences.
- The differences are:
- 13, 39, and 117.

- Notice that:
- \(39 = 3 \times 13\)
- \(117 = 9 \times 13\)

### Step 4: Predict the missing difference.
- The pattern suggests that the differences are multiples of 13, specifically odd multiples:
- \(13 \times 1\), \(13 \times 3\), and \(13 \times 9\).

- The next odd multiple of 13 would be:
- \(13 \times 5 = 65\)

### Step 5: Calculate the missing term.
- Now, add the predicted difference (65) to the last known term (58):
- \(58 + 65 = 123\)

### Step 6: Verify the next term.
- Now, check if adding the next difference (91) gives us the correct next term:
- \(123 + 91 = 214\) (which is correct)

### Conclusion:
- The missing number in the series is **123**.