Finding the Missing Number in the Series: 101 110 121 134 149 ?

The problem asks us to identify the pattern in the given number series and determine the value that should replace the question mark (?). The series provided is: 101, 110, 121, 134, 149, ?

Analyzing the Series Pattern

To find the pattern, let's examine the differences between consecutive terms in the series:

• The difference between the second term (110) and the first term (101) is: $110 - 101 = 9$
• The difference between the third term (121) and the second term (110) is: $121 - 110 = 11$
• The difference between the fourth term (134) and the third term (121) is: $134 - 121 = 13$
• The difference between the fifth term (149) and the fourth term (134) is: $149 - 134 = 15$

Identifying the Pattern in Differences

The differences we found are 9, 11, 13, and 15. We can observe a clear pattern here:

• Each difference is an odd number.
• The differences are increasing by 2 each time ($11 = 9 + 2$, $13 = 11 + 2$, $15 = 13 + 2$). This sequence of differences forms an arithmetic progression.

Calculating the Next Term

Following this pattern, the next difference in the sequence should be 2 more than the previous difference (15).

• Next difference = $15 + 2 = 17$

To find the missing term (the number that should come in place of the question mark), we add this next difference (17) to the last known term of the series (149).

• Missing term = Last term + Next difference
• Missing term = $149 + 17$
• Missing term = $166$

Conclusion

Therefore, the number that should come in place of the question mark (?) in the given series is 166.