A number series of perfect squares is given with one term missing. Choose the correct option that completes the sequence: 4, 9, 16, 25, 36, ?

Introduction / Context:
This question tests the ability to recognize a simple pattern in a number series based on perfect squares of consecutive integers. Such questions are common in competitive examinations because they quickly reveal whether a candidate can identify basic numeric structures and extend them correctly. Understanding square numbers and how they grow as the base integer increases is an important foundation for many other quantitative topics.

Given Data / Assumptions:
The series given in the question is:
4, 9, 16, 25, 36, ?
We must assume that there is a consistent rule generating each term from a simple mathematical concept, and we are expected to find the next number that follows the same rule.

Concept / Approach:
The concept involved is that of perfect squares. A perfect square is a number that can be written as n^2 for some integer n. When we look at the terms, each appears to be the square of a consecutive integer: 2^2, 3^2, 4^2, 5^2, and 6^2. Once we confirm that pattern, we can extend it by taking the next integer in order and squaring it.

Step-by-Step Solution:
Step 1: Express each term as a square of an integer.4 = 2^29 = 3^216 = 4^225 = 5^236 = 6^2Step 2: Observe that the bases are consecutive integers: 2, 3, 4, 5, 6.Step 3: The next integer after 6 is 7, so the next term should be 7^2.Step 4: Calculate 7^2.7^2 = 49.

Verification / Alternative check:
An alternative way to check is to look at the differences between the terms. The differences are 9 - 4 = 5, 16 - 9 = 7, 25 - 16 = 9, and 36 - 25 = 11. These differences themselves form a sequence of consecutive odd numbers. The difference between 36 and the next term should then be the next odd number, 13. Adding 13 to 36 also gives 49. This confirms the result from the square pattern.

Why Other Options Are Wrong:
56 is not a perfect square and does not fit the odd difference pattern. 64 is a perfect square, but it would correspond to 8^2 and would skip 7^2, breaking the sequence of consecutive squares. 21 and 94 are neither perfect squares nor consistent with the difference pattern, so they cannot be correct.

Common Pitfalls:
Candidates sometimes look only at differences and may miscalculate them, or they may incorrectly jump to 64 because it is a well known square without checking that 49 is the correct next step. Another mistake is to assume a linear pattern when the growth clearly matches the pattern of n^2. Always verify the pattern across all terms, not just the last two.

Final Answer:
The missing term in the series 4, 9, 16, 25, 36, ? is 49.