In the following number classification question, four numbers are given (26, 50, 82 and 120). In three of them, the number can be written as n^2 + 1 for an odd integer n, while one number follows a different rule. Select the odd number from the given alternatives.

Introduction / Context:
This question is from the odd one out and number pattern topic. The numbers 26, 50, 82 and 120 appear unrelated, but they are actually close to perfect squares. By examining how each number relates to the square of a nearby odd integer, you can find that three follow the pattern n^2 + 1, while one does not. That non matching number is the required odd one out.

Given Data / Assumptions:

• Numbers: 26, 50, 82 and 120.
• We consider nearby perfect squares of odd integers.
• Perfect squares of odd numbers: 5^2 = 25, 7^2 = 49, 9^2 = 81, 11^2 = 121 and so on.
• Three numbers equal square of an odd number plus 1.
• One number equals square of an odd number minus 1.

Concept / Approach:
The strategy is to compare each given number with nearby perfect squares and see whether it is one more or one less than a square. In many reasoning questions, a series is built from expressions like n^2 + 1 or n^2 - 1 with n increasing in steps. Here, 26, 50 and 82 fit n^2 + 1 for n = 5, 7 and 9 respectively, whereas 120 fits 11^2 - 1. Recognising this contrast leads directly to the correct answer.

Step-by-Step Solution:
Step 1: Compare 26 with nearby squares.5^2 = 25, so 25 + 1 = 26. Thus 26 = 5^2 + 1.Step 2: Compare 50 with nearby squares.7^2 = 49, so 49 + 1 = 50. Thus 50 = 7^2 + 1.Step 3: Compare 82 with nearby squares.9^2 = 81, so 81 + 1 = 82. Thus 82 = 9^2 + 1.Step 4: Compare 120 with nearby squares.11^2 = 121. Now 121 - 1 = 120, so 120 = 11^2 - 1, not 11^2 + 1.Step 5: Identify the odd number.Three numbers fit the rule n^2 + 1 for odd n, but 120 fits n^2 - 1. Therefore, 120 is the odd one out.

Verification / Alternative check:
To verify, you can list the sequence of n^2 + 1 for n = 5, 7, 9 and 11. You get 26, 50, 82 and 122. Among the given options, 26, 50 and 82 appear in this list, but 120 does not. Instead, 120 appears in the sequence n^2 - 1 for n = 11, which gives 121 - 1 = 120. This confirms that 120 belongs to a different pattern than the other three numbers.

Why Other Options Are Wrong:
26: Equal to 5^2 + 1, matches the main pattern.
50: Equal to 7^2 + 1, again following the same rule.
82: Equal to 9^2 + 1, also consistent with the n^2 + 1 structure.

Common Pitfalls:
Many students first attempt to look at differences between numbers or factors, which can make the pattern hard to see. When numbers are close to well known squares like 25, 49, 81 or 121, a better first step is to check whether they are one more or one less than these squares. Remembering squares of odd integers up to at least 15^2 gives you a strong advantage in detecting such patterns quickly.

Final Answer:
The odd number is 120, because 26, 50 and 82 are of the form n^2 + 1 for odd n, whereas 120 is of the form n^2 - 1 for n = 11.