The question asks us to identify the missing number in the given series: 5, 10, 17, 26, 37, ?, 65. To find the missing number, we need to carefully analyze the pattern or rule that connects the terms in this number series.

Number Series Pattern Analysis

Let's examine the differences between consecutive terms in the given number series. This often reveals an underlying pattern, especially in arithmetic or quadratic series.

• The difference between the second term (10) and the first term (5) is $10 - 5 = 5$.
• The difference between the third term (17) and the second term (10) is $17 - 10 = 7$.
• The difference between the fourth term (26) and the third term (17) is $26 - 17 = 9$.
• The difference between the fifth term (37) and the fourth term (26) is $37 - 26 = 11$.

Let's present these differences in a clear table to observe the pattern more easily:

Solving the Missing Number Series

Upon observing the differences (5, 7, 9, 11), we can see a clear pattern: these differences are consecutive odd numbers. Each difference is 2 more than the previous one. This indicates that the series is a quadratic progression, where the second level of differences is constant.

• The next difference in this sequence of odd numbers should be $11 + 2 = 13$.

Therefore, to find the missing number, we must add this next difference (13) to the last known term (37):

Missing number $= 37 + 13 = 50$.

To verify this, let's see if the next term (65) fits the pattern. The difference after 13 should be $13 + 2 = 15$.

If the missing number is 50, then the difference between 65 and 50 should be 15:

$65 - 50 = 15$.

Since this matches our expected difference, our calculated missing number of 50 is correct.

Alternate Method for Series Pattern

Another way to identify the pattern in this number series is by looking at how each term relates to perfect squares:

• First term: $5 = 2^2 + 1$
• Second term: $10 = 3^2 + 1$
• Third term: $17 = 4^2 + 1$
• Fourth term: $26 = 5^2 + 1$
• Fifth term: $37 = 6^2 + 1$

Following this pattern, the missing term (which is the sixth term) should be:

• Sixth term: $7^2 + 1 = 49 + 1 = 50$.

And the seventh term (65) confirms this pattern:

• Seventh term: $8^2 + 1 = 64 + 1 = 65$.

Both methods consistently lead to the same missing number, confirming the accuracy of our solution.

Final Missing Number

Based on the consistent pattern found through both difference analysis and square relationship, the missing number in the series 5, 10, 17, 26, 37, ?, 65 is 50.