Find the odd term out of the series 4, 5, 9, 18, 32, 59.

Introduction / Context:

This question provides a moderately increasing number series and asks for the odd term out, that is, the term that breaks the rule followed by the others. The series appears to involve incremental additions that may be based on squares of integers. Identifying that structure is the key to solving the problem correctly.

Given Data / Assumptions:

• The series is 4, 5, 9, 18, 32, 59.
• Exactly one term is inconsistent with the underlying pattern.
• The incremental jumps between numbers suggest they might be related to perfect squares.
• We must find which number does not fit that pattern.

Concept / Approach:

We first calculate the differences between consecutive terms and see whether those differences themselves follow a clear rule. In this series, the differences appear close to 1^2, 2^2, 3^2, 4^2, 5^2. If we can confirm that idea by reconstructing the series using these squares, and if one term does not fit, that term must be the odd one out. This method of looking at difference sequences is very powerful in many number series questions.

Step-by-Step Solution:

Step 1: Compute consecutive differences: 5 - 4 = 1, 9 - 5 = 4, 18 - 9 = 9, 32 - 18 = 14, 59 - 32 = 27. Step 2: Compare these with squares: 1, 4, 9, 16, 25 are 1^2, 2^2, 3^2, 4^2, 5^2. Step 3: The differences 1, 4, 9, and 27 match 1^2, 2^2, 3^2, and 5^2 fairly closely, but the difference 14 does not match 4^2, which should be 16. Step 4: Construct the ideal series by adding successive squares: start with 4, add 1 to get 5, add 4 to get 9, add 9 to get 18, add 16 to get 34, and add 25 to get 59. Step 5: The ideal sequence is thus 4, 5, 9, 18, 34, 59. Comparing this with the given series, 32 is the only term that differs from the intended 34, so 32 is the odd term.

Verification / Alternative check:

Using the rule add n^2 for n from 1 onwards, we have: 4 + 1^2 = 5, 5 + 2^2 = 9, 9 + 3^2 = 18, 18 + 4^2 = 34, 34 + 5^2 = 59. Every step in this reconstruction uses a perfect square in ascending order. The resulting sequence is consistent and elegant, which is exactly the type of pattern examination questions are designed around. The given sequence deviates from this pattern only at the fifth term, where 32 is provided instead of 34.

Why Other Options Are Wrong:

Term 5 is obtained correctly from 4 by adding 1^2. Term 9 correctly follows from 5 by adding 2^2. Term 18 is exactly 9 plus 3^2. Term 59 is 34 plus 5^2 in the ideal series, and only differs through the error in the preceding term. Removing 59 would break the neat use of 5^2 as the final increment. Therefore, 32 is uniquely inconsistent with the rule of adding successive perfect squares, and the other options are necessary to maintain the pattern.

Common Pitfalls:

One error is to miscalculate squares or differences, which easily leads to confusion. Another is to assume the odd term must be at the end of the series instead of checking all positions carefully. Some learners also attempt to find multiplicative relations when the series is clearly built from additive increments. Focusing on how differences relate to basic sequences such as squares or cubes is often the best path to the solution.

Final Answer:

The term that breaks the pattern of adding consecutive perfect squares is 32, so 32 is the odd term out.