Choose the correct alternative that will continue the same number pattern and replace the missing term x in the series 4, 9, 25, x, 121, 169, 289, 361.

Introduction / Context:

This question tests number series reasoning where each term follows a hidden mathematical rule. Learners must identify the pattern across the entire sequence, not just locally between two numbers. The focus here is on recognizing perfect squares linked with prime numbers and then using that pattern to determine the missing term x. Such questions appear very frequently in aptitude tests, banking examinations, and other competitive exams where numerical logic and observation skills are evaluated.

Given Data / Assumptions:

• The given number series is 4, 9, 25, x, 121, 169, 289, 361.
• Exactly one term x is missing, and we must replace it using the same pattern that generates the other terms.
• All terms are positive integers and appear to be perfect squares.
• Only one of the given answer options will fit the underlying rule consistently.

Concept / Approach:

The best way to approach this series is to look at the terms as possible squares of integers, and then see if those integers follow some special pattern. If we can express each term as n^2, the base numbers may follow a sequence like natural numbers, prime numbers, odd numbers, or some other standard pattern. Once we identify the pattern in the base numbers, we can square the missing base number to get the required value of x. This approach converts the original series problem into a simpler pattern recognition task on smaller integers.

Step-by-Step Solution:

Step 1: Express each visible term as a square: 4 = 2^2, 9 = 3^2, 25 = 5^2, 121 = 11^2, 169 = 13^2, 289 = 17^2, 361 = 19^2. Step 2: Look at the base numbers: 2, 3, 5, ?, 11, 13, 17, 19. Step 3: Observe that these base numbers are prime numbers in ascending order: 2, 3, 5, 7, 11, 13, 17, 19. Step 4: The missing prime between 5 and 11 is 7. Therefore the missing term x must be 7^2. Step 5: Compute 7^2 = 49. So x = 49 is the only value that preserves the prime square pattern.

Verification / Alternative check:

To verify, rewrite the entire series using the identified rule: 2^2 = 4, 3^2 = 9, 5^2 = 25, 7^2 = 49, 11^2 = 121, 13^2 = 169, 17^2 = 289, 19^2 = 361. The base sequence 2, 3, 5, 7, 11, 13, 17, 19 is exactly the sequence of prime numbers from 2 to 19. There are no breaks or inconsistencies when we choose 49 as the missing term. Any other number in that position would destroy this elegant prime square pattern. This confirms that our interpretation of the series is mathematically consistent and robust.

Why Other Options Are Wrong:

Option 81 is 9^2, but 9 is not a prime, so it breaks the prime number pattern. Option 36 is 6^2, and 6 is composite and also not in the observed prime sequence. Option 64 is 8^2, and 8 is again not a prime. Option 100 is 10^2, but 10 is composite and does not fit into the strict prime sequence between 5 and 11. Therefore none of these alternatives maintain the consistent structure of squares of consecutive prime numbers.

Common Pitfalls:

A common mistake is to look only at the differences between consecutive terms and try to force a pattern with additions or subtractions, which in this case looks messy and non uniform. Another error is to assume the squares relate to ordinary consecutive integers instead of prime numbers. Some students also get misled by large numbers like 289 and 361 and stop checking whether they are squares of primes. Ignoring the nature of the base numbers and focusing only on raw values often leads to incorrect choices such as 36 or 64.

Final Answer:

The missing term that maintains the pattern of squares of consecutive prime numbers is 49, so the correct answer is 49.