Number Series Analysis: Finding the Missing Term

The question presents a sequence of numbers: 57, 63, 71, ?, 93, 107. We need to identify the pattern governing this series to find the missing number represented by the question mark (?).

Pattern Discovery in the Series

To find the pattern, let's look at the differences between consecutive terms:

• The difference between the second term (63) and the first term (57) is $63 - 57 = 6$.
• The difference between the third term (71) and the second term (63) is $71 - 63 = 8$.

Observing these differences (6, 8), we can see that they are increasing. The increase between the first two differences is $8 - 6 = 2$. This suggests that the differences might be consecutive even numbers, increasing by 2 each time.

Let's test this hypothesis:

• The next difference should be $8 + 2 = 10$.
• The difference after that should be $10 + 2 = 12$.
• The final difference should be $12 + 2 = 14$.

This sequence of differences is 6, 8, 10, 12, 14.

Step-by-Step Solution

Using the identified pattern of increasing differences, we can find the missing number:

1. Start with the first term: 57.
2. Add the first difference: $57 + 6 = 63$. (Matches the second term)
3. Add the second difference: $63 + 8 = 71$. (Matches the third term)
4. Add the third difference (which we predicted as 10): $71 + 10 = 81$. This is the potential missing number.
5. Add the fourth difference (predicted as 12): $81 + 12 = 93$. (Matches the fifth term)
6. Add the fifth difference (predicted as 14): $93 + 14 = 107$. (Matches the sixth term)

The pattern holds true for the entire series when we assume the missing number is 81.

Conclusion

The sequence follows a pattern where we add consecutive even integers starting from 6 to the previous term. Therefore, the number that replaces the question mark (?) is 81.