A number series is given with one term missing. Study the pattern in 1, 0.25, 1/9, ?, 0.04, 1/36 and choose the correct alternative that will complete the series.

Introduction / Context:
This number series uses reciprocals of perfect squares. The given sequence is 1, 0.25, 1/9, ?, 0.04, 1/36. Our task is to detect the pattern that links each term and then determine the missing value. Questions of this type test knowledge of basic squares and the ability to recognise when a series is built from them in decimal or fractional form.

Given Data / Assumptions:
Given series: 1, 0.25, 1/9, ?, 0.04, 1/36.
Some terms are decimals, others are written as fractions.
We assume the terms are related to each other by a simple mathematical formula.
Exactly one of the provided options will match the required missing term.

Concept / Approach:
A sensible approach is to convert every term to a single representation, for example as reciprocals of squares. 1 can be written as 1/1^2, 0.25 as 1/4, which is 1/2^2, 1/9 is already in the form 1/3^2, 0.04 is 1/25, that is 1/5^2, and 1/36 is 1/6^2. This strongly suggests that the series is built from 1/(n^2) for n = 1, 2, 3, 4, 5, 6. Therefore, the missing term should be 1/4^2 = 1/16, written either as a fraction or as a decimal.

Step-by-Step Solution:
Express each term as 1/(n^2): 1 = 1/1^2. 0.25 = 1/4 = 1/2^2. 1/9 = 1/3^2. The missing term should be 1/4^2 = 1/16. 0.04 = 1/25 = 1/5^2. 1/36 = 1/6^2. 1/16 as a decimal is 0.0625. Hence, the missing term is 0.0625.

Verification / Alternative check:
We list the denominators of the reciprocals: 1^2, 2^2, 3^2, 4^2, 5^2, 6^2. The sequence of n values is 1, 2, 3, 4, 5, 6, so the series is clearly built from consecutive positive integers. All given terms match this rule except the missing one, which must correspond to n = 4. Therefore 1/16 or 0.0625 is the only value that fits the pattern seamlessly.

Why Other Options Are Wrong:
Option B: 0.025 equals 1/40 and does not correspond to 1/4^2 or any 1/n^2 with small integer n in this context.
Option C: 1/8 is not the reciprocal of a perfect square but of 2^3, so it is inconsistent with the pattern.
Option D: 1/6 is also not of the form 1/n^2 for n = 1 to 6 in this series.
Option E: 0.01 equals 1/100, which would require n = 10, outside the small consecutive range implied here.

Common Pitfalls:
Students often overlook that decimals like 0.25 and 0.04 correspond to very clean fractional forms related to squares. Another pitfall is to search for an arithmetic progression in the decimal values instead of spotting the underlying 1/n^2 structure. Strong familiarity with squares from 1^2 up to at least 20^2 is extremely useful for quickly solving such problems in exams.

Final Answer:
The missing term that keeps the sequence of reciprocals of consecutive squares intact is 0.0625, so the correct option is 0.0625.