Analyzing the Number Series Pattern

The question asks us to find the missing number in the given numerical series: 37, 41, 50, 66, ?. To solve this, we need to identify the pattern governing the sequence. Let's look at the differences between consecutive terms:

• The difference between the second term (41) and the first term (37) is: $41 - 37 = 4$
• The difference between the third term (50) and the second term (41) is: $50 - 41 = 9$
• The difference between the fourth term (66) and the third term (50) is: $66 - 50 = 16$

Identifying the Pattern of Differences

Now, let's examine the differences we found: 4, 9, 16. We can observe that these numbers are perfect squares:

• $4 = 2 \times 2 = 2^2$
• $9 = 3 \times 3 = 3^2$
• $16 = 4 \times 4 = 4^2$

The pattern shows that the difference between consecutive terms is the square of consecutive integers starting from 2.

Calculating the Missing Number

Following this pattern, the next difference should be the square of the next integer, which is 5.

• The next difference is $5^2 = 5 \times 5 = 25$.

To find the missing number in the series, we add this next difference (25) to the last known term (66):

Missing Number = Last Term + Next Difference

Missing Number = $66 + 25$

Missing Number = $91$

Therefore, the number that replaces the question mark (?) in the series is 91.