Choose the correct alternative that will continue the same pattern and replace x in the series: 3, 12, 27, 48, 75, x, 147.

Introduction / Context:
This number series problem is based on a simple formula involving squares of natural numbers. Each term follows a regular pattern that can be expressed as a multiple of the square of its position in the series. Recognising when numbers relate to squares or cubes is a key skill in quantitative aptitude questions.

Given Data / Assumptions:

• Series: 3, 12, 27, 48, 75, x, 147.
• One term x is missing before the final given term.
• The pattern is likely related to squares or another familiar function of the position in the series.

Concept / Approach:
When terms increase at a rate that roughly follows n^2, n^3, or similar growth, it is useful to check if each term can be written in terms of those powers. Here, every term can be expressed as 3 times a perfect square. Once this pattern is confirmed, we can generate the missing term by applying the same rule to the next position in the sequence.

Step-by-Step Solution:
Step 1: Label the positions of the series terms: position 1 has value 3, position 2 has 12, position 3 has 27, position 4 has 48, position 5 has 75, position 6 is x, and position 7 has 147. Step 2: Express each known term in terms of its position. At position 1: 3 = 3 * 1^2. At position 2: 12 = 3 * 2^2. At position 3: 27 = 3 * 3^2. At position 4: 48 = 3 * 4^2. At position 5: 75 = 3 * 5^2. At position 7: 147 = 3 * 7^2. Step 3: The pattern is clearly term(n) = 3 * n^2. Step 4: For position 6, compute term(6) = 3 * 6^2 = 3 * 36 = 108.

Verification / Alternative check:
List the full series using the formula term(n) = 3 * n^2: for n = 1 to 7, we have 3, 12, 27, 48, 75, 108, 147. This exactly reproduces all the given terms and fills in the missing one. There are no inconsistencies, which confirms that the rule is correct and that the missing value must be 108.

Why Other Options Are Wrong:
Values 127, 112, or 85 are not equal to 3 times any perfect square. For example, 127 is not 3 * k^2 for an integer k. Substituting any of these numbers would break the neat formula that holds for all other terms in the series. Therefore these options cannot be correct under the identified pattern term(n) = 3 * n^2.

Common Pitfalls:
Some candidates try to work with differences between consecutive terms, which here are 9, 15, 21, 27, ... and also follow a pattern, but this can be slightly more cumbersome. Recognising the connection to squares often provides a faster route. Another common mistake is to assume a random rule based only on the first few terms without checking the entire series including the last fixed term, which is crucial here.

Final Answer:
The value of x that correctly completes the series is 108.