A series is given with one term missing. Select the correct alternative that will complete the series: 4, 9, 25, 64, 169, ?

Introduction / Context:
This question involves a number series based on perfect squares. However, the interesting twist is that the bases of these squares themselves follow a well known pattern. Questions like this test your ability to recognize both square numbers and familiar sequences such as the Fibonacci series, which is common in aptitude and competitive exams.

Given Data / Assumptions:

• Series: 4, 9, 25, 64, 169, ?
• Exactly one term is missing at the end.
• Each visible term appears to be a perfect square.

Concept / Approach:
First, identify whether the given numbers are perfect squares. If they are, then look at the square roots and see if those roots follow a simpler sequence. In this question, those roots form a variant of the Fibonacci sequence. Once that underlying pattern is recognised, you can square the next base term to obtain the missing number in the original series.

Step-by-Step Solution:
Step 1: Write each number as a square if possible. 4 = 2^2. 9 = 3^2. 25 = 5^2. 64 = 8^2. 169 = 13^2. Step 2: Collect the square roots: 2, 3, 5, 8, 13. Step 3: Notice that these roots follow a Fibonacci type rule, where each term is approximately the sum of the previous two: 2, 3, 5, 8, 13, 21, ... Step 4: The next base should therefore be 21, the next Fibonacci number after 13. Step 5: Square this base: 21^2 = 441.

Verification / Alternative check:
Extend the list of squares using the Fibonacci based roots: 2^2 = 4, 3^2 = 9, 5^2 = 25, 8^2 = 64, 13^2 = 169, 21^2 = 441. This exactly matches the given series and adds the missing term. There is no simpler alternative that connects all the given numbers as elegantly as this Fibonacci squared pattern, so the result is strongly validated.

Why Other Options Are Wrong:
225 is 15^2 and 256 is 16^2, while 289 is 17^2. None of the bases 15, 16, or 17 fit into the Fibonacci root pattern 2, 3, 5, 8, 13, 21. Choosing any of these would break the relationship where each new base is formed from the sum of the two previous bases, which is the key idea behind this question.

Common Pitfalls:
Many candidates stop after noticing that the numbers are squares and then try to find a pattern directly among 4, 9, 25, 64, 169 without looking at the roots. Others may try to impose a complicated difference pattern. The more reliable strategy is to move one level down to the square roots and check whether they follow a familiar sequence like the Fibonacci series.

Final Answer:
The missing term in the series is 441.