37, ?, 103, 169, 257, 367

Options

A. 61
B. 59
C. 67
D. 55

Explanation

To find the missing number in the series 37, ?, 103, 169, 257, 367, we will analyze the differences between the consecutive numbers.

### Step-by-Step Solution:

1. **Identify the given series**:
The series is: 37, ?, 103, 169, 257, 367.

2. **Calculate the differences between known consecutive numbers**:
- Difference between 169 and 103:
\[ 169 - 103 = 66 \]
- Difference between 257 and 169:
\[ 257 - 169 = 88 \]
- Difference between 367 and 257:
\[ 367 - 257 = 110 \]

3. **List the differences**:
The differences we have calculated are:
- From 103 to 169: 66
- From 169 to 257: 88
- From 257 to 367: 110

4. **Find the pattern in the differences**:
Now, let's find the differences between the differences:
- Difference between 88 and 66:
\[ 88 - 66 = 22 \]
- Difference between 110 and 88:
\[ 110 - 88 = 22 \]
This shows that the differences are increasing by 22.

5. **Calculate the missing difference**:
If we continue this pattern backwards, the difference before 66 should be:
\[ 66 - 22 = 44 \]

6. **Find the missing number**:
Now, we can find the missing number by subtracting the last known difference (44) from 103:
\[ 103 - 44 = 59 \]

7. **Final answer**:
Therefore, the missing number in the series is **59**.

### Verification:
To ensure our solution is correct, we can check the differences:
- From 37 to 59:
\[ 59 - 37 = 22 \]
- From 59 to 103:
\[ 103 - 59 = 44 \]
- From 103 to 169:
\[ 169 - 103 = 66 \]
- From 169 to 257:
\[ 257 - 169 = 88 \]
- From 257 to 367:
\[ 367 - 257 = 110 \]

The differences are consistent with our earlier calculations, confirming that our missing number is indeed correct.