A number series of perfect squares is given with one term missing. Choose the correct alternative from the options that will complete the series: 49, 64, ?, 100, 121.

Introduction / Context:
This question presents a number series made of perfect squares. The task is to identify which square is missing and select it from the options. Recognizing perfect squares and their corresponding integer roots is a fundamental number skill used often in aptitude and reasoning questions.

Given Data / Assumptions:
Given series: 49, 64, ?, 100, 121.We must find the missing third term.We assume the numbers are consecutive squares of natural numbers in increasing order.

Concept / Approach:
Perfect squares are numbers of the form n^2, where n is an integer. To solve this question, we identify the integer roots of the known terms and check whether they follow a consecutive sequence. Once we see which integer is missing in that sequence, we square it to obtain the missing term. This method is simple and reliable for square-based series.

Step-by-Step Solution:
Step 1: Express 49 as a square: 49 = 7^2.Step 2: Express 64 as a square: 64 = 8^2.Step 3: Express 100 as a square: 100 = 10^2.Step 4: Express 121 as a square: 121 = 11^2.Step 5: The sequence of roots is 7, 8, ?, 10, 11. It is clear that 9 is missing between 8 and 10.Step 6: Square the missing root: 9^2 = 81, so the missing term is 81.

Verification / Alternative check:
Write the completed series of squares: 49 (7^2), 64 (8^2), 81 (9^2), 100 (10^2), 121 (11^2). The roots form a clean sequence of consecutive integers from 7 to 11. This is a typical exam pattern and confirms that the series is indeed built from consecutive square numbers. Since 81 is the only value that completes this structure, it must be the correct answer.

Why Other Options Are Wrong:
Option 74 is not a perfect square and does not correspond to any integer root.Option 80 is also not a perfect square; its square root is not an integer.Option 75 is similarly non-square and would break the pattern of consecutive perfect squares.

Common Pitfalls:
Some learners may attempt to look at differences between terms rather than recognizing the square pattern, which can be misleading here. Another mistake is confusing near-square numbers such as 80 or 75 with actual squares. Memorizing squares of numbers from 1 to at least 20 is extremely helpful for quickly identifying such patterns in exams.

Final Answer:
The missing term that completes the series of perfect squares is 81.