Number Series: Finding the Missing Term

The task is to determine the missing number in the sequence: 15, 22, 33, 50, 71, 98, ?. We need to analyze the relationship between the numbers to uncover the pattern.

Analyzing the Series Pattern

First, let's calculate the differences between consecutive terms:

• $22 - 15 = 7$
• $33 - 22 = 11$
• $50 - 33 = 17$
• $71 - 50 = 21$
• $98 - 71 = 27$

The differences obtained are: 7, 11, 17, 21, 27.

Identifying the Pattern in Differences

Let's examine the differences between these consecutive differences (the second differences):

• $11 - 7 = 4$
• $17 - 11 = 6$
• $21 - 17 = 4$
• $27 - 21 = 6$

We can see a clear alternating pattern in the second differences: 4, 6, 4, 6.

Calculating the Next Term

To find the next term in the original series, we first need to determine the next difference. Based on the established pattern (4, 6, 4, 6), the next number to add in the second difference sequence is 4.

• The next difference in the first sequence (7, 11, 17, 21, 27) will be the last difference plus 4: $27 + 4 = 31$.
• Now, we add this calculated difference (31) to the last term of the original series (98) to find the missing term: $98 + 31 = 129$.

Conclusion

The number series follows a pattern where the difference between consecutive terms increases by alternating values of 4 and 6. Following this logic, the missing number in the series 15, 22, 33, 50, 71, 98 is 129.