What will come in place of question mark (?) in the given number series?
5 9 33 72 121 ?

Options

A. 163
B. 171
C. 184
D. 175

Answer

To find the missing number in the series 5, 9, 33, 72, 121, ?, we will analyze the differences between consecutive terms.

1. **Identify the series**:
The given series is: 5, 9, 33, 72, 121, ?

2. **Calculate the differences between consecutive terms**:
- Difference between 9 and 5:
\( 9 - 5 = 4 \)
- Difference between 33 and 9:
\( 33 - 9 = 24 \)
- Difference between 72 and 33:
\( 72 - 33 = 39 \)
- Difference between 121 and 72:
\( 121 - 72 = 49 \)

So, the differences we have calculated are: 4, 24, 39, 49.

3. **Calculate the differences of the differences**:
- Difference between 24 and 4:
\( 24 - 4 = 20 \)
- Difference between 39 and 24:
\( 39 - 24 = 15 \)
- Difference between 49 and 39:
\( 49 - 39 = 10 \)

The second level differences are: 20, 15, 10.

4. **Identify the pattern in the second level differences**:
The second level differences are decreasing by 5 each time:
- From 20 to 15: decrease of 5
- From 15 to 10: decrease of 5

Following this pattern, the next difference should be:
\( 10 - 5 = 5 \)

5. **Calculate the next first level difference**:
Adding this to the last first level difference:
\( 49 + 5 = 54 \)

6. **Find the missing term**:
Now, we add this difference to the last known term in the series:
\( 121 + 54 = 175 \)

Thus, the missing number in the series is **175**.