In the sequence 2, 5, 10, 17, 26, 37, 50, 64, which number is the odd one out?

Introduction / Context:
This question asks you to find the odd man out in a number series that follows a certain rule. The pattern is not a simple arithmetic progression, but instead involves increasing differences between consecutive terms. Recognizing such difference patterns is crucial in many series questions where the terms seem irregular at first glance.

Given Data / Assumptions:

• The series given is 2, 5, 10, 17, 26, 37, 50, 64.
• We need to identify a single term that does not follow the pattern that the other terms follow.
• The numbers are positive and increase as we move along the series.

Concept / Approach:
A standard technique is to look at the differences between consecutive terms. If those differences themselves follow a regular pattern, we can then test each given term to see which one violates the rule. In this series, the differences form an increasing sequence that suggests a quadratic like pattern where the difference increases by a constant amount each time. Any term that disrupts this systematic increase will be the odd one out.

Step-by-Step Solution:
Compute the differences between consecutive terms: 5 - 2 = 3. 10 - 5 = 5. 17 - 10 = 7. 26 - 17 = 9. 37 - 26 = 11. 50 - 37 = 13. 64 - 50 = 14. We observe that the differences are 3, 5, 7, 9, 11, 13, and 14. The first six differences form an odd number sequence: 3, 5, 7, 9, 11, 13. Following this pattern, the next difference should be 15, not 14. So the term after 50 should be 50 + 15 = 65 to continue the pattern correctly. Instead, we have 64 in the given series, which breaks the pattern. Therefore, 64 is the odd man out.

Verification / Alternative Check:
If we reconstruct the ideal pattern based on the rule that each difference increases by 2 starting from 3, we get differences 3, 5, 7, 9, 11, 13, 15. Starting from 2, the next terms would then be: 2 + 3 = 5, 5 + 5 = 10, 10 + 7 = 17, 17 + 9 = 26, 26 + 11 = 37, 37 + 13 = 50, and 50 + 15 = 65. Thus, the logically consistent series is 2, 5, 10, 17, 26, 37, 50, 65. This confirms that 64 does not fit and that substituting 65 would restore the pattern.

Why Other Options Are Wrong:

• Option 50: The difference from 37 to 50 is 13, fitting perfectly into the odd differences pattern.
• Option 37: The difference from 26 to 37 is 11, also following the odd number sequence.
• Option 26: The difference from 17 to 26 is 9, consistent with the pattern.
• Option 10: The difference from 5 to 10 is 5, again fitting the planned sequence.

Common Pitfalls:
One common mistake is to treat the series as random and guess without checking the differences. Another error is to miscalculate the differences and not notice the odd number pattern in them. Some students also look for relationships between nonconsecutive terms and miss the simpler idea that consecutive differences should increase by a constant amount. Always start by calculating first differences and, if needed, second differences when analyzing such series.

Final Answer:
The odd man out in the series is 64.