In the series 12, 16, 32, 68, ?, find the missing term.

Introduction / Context:
This number series problem relies on recognising a pattern in the differences between consecutive terms. The differences here turn out to follow a regular progression that grows steadily, making this a classic aptitude question.

Given Data / Assumptions:

• Series: 12, 16, 32, 68, ?
• Exactly one term is missing.
• The rule that generates each new term is consistent across the series.

Concept / Approach:
When there is no obvious multiplicative pattern, it is standard practice to look at the first-level differences. If those differences themselves follow an arithmetic progression or some other simple rule, we can extend that pattern to determine the next difference and hence the next term.

Step-by-Step Solution:
Step 1: Compute differences: 16 − 12 = 4, 32 − 16 = 16, 68 − 32 = 36. Step 2: Look at these differences: 4, 16, 36. Notice that 4 = 2^2, 16 = 4^2, 36 = 6^2. They are squares of even numbers 2, 4, and 6. Step 3: This suggests that the next difference may be 8^2 = 64, continuing the pattern of even squares 2^2, 4^2, 6^2, 8^2. Step 4: Add this difference to the last known term: 68 + 64 = 132.

Verification / Alternative check:
With the candidate term 132, the series becomes 12, 16, 32, 68, 132. The differences are 4, 16, 36, 64 which correspond to 2^2, 4^2, 6^2, 8^2. The pattern of square numbers of consecutive even integers is clean and consistent, confirming that 132 is correct.

Why Other Options Are Wrong:
Options 198, 164, and 118 yield differences that do not fit the even square pattern. For instance, if the term were 164, the last difference would be 96, which is not a perfect square. These choices would destroy the simple structure observed in the first three differences.

Common Pitfalls:
A frequently seen mistake is to search for a direct formula on the main terms and ignore the pattern in the differences. Another error is to assume the differences are random or only linear, whereas in many exam series they are derived from squares or cubes. Taking the time to check for perfect squares and cubes in differences often reveals the intended pattern quickly.

Final Answer:
The missing term that preserves the even square difference pattern is 132.