A series is given with one term missing. Choose the correct alternative from the given options that will complete the series: 144, 256, 400, ?

Introduction / Context:
This question is built around perfect squares of specific integers. The series 144, 256, 400, ? consists of well known square numbers and tests whether candidates can recognise the pattern of squaring consecutive multiples of a fixed integer. We must determine the correct next square in this pattern.

Given Data / Assumptions:
- Known terms: 144, 256, 400.- One term is missing after 400.- All numbers are perfect squares.- The bases of these squares are likely to follow a simple pattern, such as equally spaced multiples.

Concept / Approach:
We rewrite each term as a square: n^2. By determining the sequence of n values, we can identify the next n and square it to get the missing term. Recognising common square values like 12^2, 16^2 and 20^2 is the key step here.

Step-by-Step Solution:
- 144 = 12^2.- 256 = 16^2.- 400 = 20^2.- The bases are 12, 16 and 20, which are multiples of 4: 4 * 3, 4 * 4 and 4 * 5.- These bases increase by 4 each time: 12, 16, 20, so the next base should be 24.- Compute 24^2: 24 * 24 = 576.- Therefore the missing term in the series is 576.

Verification / Alternative check:
- Sequence of square bases: 12, 16, 20, 24 are consecutive multiples of 4.- Their squares are 144, 256, 400, 576.- Thus the complete series of terms is 144, 256, 400, 576, which fits a clear and simple pattern.

Why Other Options Are Wrong:
- 441 = 21^2 and 289 = 17^2, which are not based on the multiples of 4 that we observe.- 625 = 25^2 and 512 is not a perfect square at all; both disrupt the neat sequence of square numbers of 12, 16, 20, 24.- None of these alternatives preserves the logic of using consecutive multiples of 4 as bases.

Common Pitfalls:
- Some candidates may focus only on differences between terms, which are 112 and 144, and fail to see the square pattern.- Mistaking 512 for a square number can lead to incorrect choices.- Forgetting standard square values such as 24^2 can result in unnecessary calculations or guesswork.

Final Answer:
The terms are squares of consecutive multiples of 4 (12, 16, 20, 24), so the missing number is 576.