A series is given, with one number missing. Choose the correct alternative that will complete the series of reciprocals of perfect cubes: 1, 0.125, 1/27, 1/64, ?, 1/216.

Introduction / Context:
This number series involves both fractions and decimal representations of reciprocals. The goal is to detect the pattern of denominators and identify the missing term. Such questions test familiarity with cubes and their reciprocals, as well as the ability to recognise equivalences between decimals and fractions.

Given Data / Assumptions:

- The given series is 1, 0.125, 1/27, 1/64, ?, 1/216.- 0.125 is equal to 1/8 in fractional form.- The missing term is between 1/64 and 1/216.

Concept / Approach:
The denominators 1, 8, 27, 64 and 216 are all familiar perfect cubes: 1^3, 2^3, 3^3, 4^3 and 6^3. It is natural to suspect that the series lists reciprocals of consecutive cubes. If that is the rule, the missing term should correspond to the cube that lies between 4^3 and 6^3, which is 5^3. We must then match the reciprocal of 5^3 with one of the options, possibly in decimal form.

Step-by-Step Solution:
Step 1: Rewrite the numbers in fractional form where needed. 1 = 1/1 and 0.125 = 1/8.Step 2: Identify denominators: 1, 8, 27, 64, ?, 216.Step 3: Recognise that 1 = 1^3, 8 = 2^3, 27 = 3^3, 64 = 4^3 and 216 = 6^3.Step 4: The sequence of cube bases is 1, 2, 3, 4, ?, 6, so the missing base is 5.Step 5: Compute 5^3 = 125, so the missing term should be 1/125.Step 6: Convert 1/125 to decimal form: 1 / 125 = 0.008.Step 7: Therefore, the missing term is 0.008, which represents 1/125.

Verification / Alternative check:
List the full series in a uniform form: 1/1, 1/8, 1/27, 1/64, 1/125, 1/216. The denominators now are clearly 1^3, 2^3, 3^3, 4^3, 5^3 and 6^3. This proves that we have reciprocals of consecutive cubes from 1 to 6, confirming that 1/125 or 0.008 is the correct missing term.

Why Other Options Are Wrong:
Option 0.025 equals 1/40, which is not a cube denominator. Option 1/8 repeats the earlier term and breaks the pattern of distinct denominators. Option 1/128 uses 128, which is not the cube of an integer. Hence none of these values fit the rule of reciprocals of consecutive perfect cubes.

Common Pitfalls:
Candidates sometimes focus only on the decimal values and miss the underlying structure in the denominators. Another common mistake is to treat 0.125 and 0.008 as unrelated decimals instead of converting them to fractions. When a series involves fractions or decimals, always check whether the denominators form a simple pattern like consecutive squares or cubes.

Final Answer:
The missing number that correctly completes the series is 0.008 (which equals 1/125).