In the following question, select the missing number from the given series: 3, 8, 5, 27, 8, 64, 12, 125, 17, ?

Introduction / Context:
This series can be better understood by grouping the numbers into pairs. Each pair consists of a smaller first term and a much larger second term. A common pattern in such cases is that the second term of each pair is a cube, while the first terms follow a separate additive rule. Recognising the two interwoven sequences is the key to solving this question correctly.

Given Data / Assumptions:

• Series: 3, 8, 5, 27, 8, 64, 12, 125, 17, ?
• Consider the series as pairs: (3, 8), (5, 27), (8, 64), (12, 125), (17, ?).
• The second elements look like perfect cubes.

Concept / Approach:
Split the series into two subsequences. One subsequence uses the first number of each pair, and the other uses the second number of each pair. Often the second numbers form powers like cubes or squares, while the first numbers form a simple additive progression. Once we identify both patterns, we can fill in the missing value in the final pair.

Step-by-Step Solution:
Step 1: Form pairs: (3, 8), (5, 27), (8, 64), (12, 125), (17, ?). Step 2: Examine the second elements: 8, 27, 64, 125. These are 2^3, 3^3, 4^3, and 5^3 respectively. Step 3: This suggests that the series of second elements is cubes of consecutive integers: 2^3, 3^3, 4^3, 5^3, 6^3. Step 4: The next cube should be 6^3 = 216. Step 5: Now examine the first elements of each pair: 3, 5, 8, 12, 17. Differences between these first elements are: 5 - 3 = 2, 8 - 5 = 3, 12 - 8 = 4, 17 - 12 = 5. Step 6: The first elements increase by 2, then 3, then 4, then 5, which is a simple pattern of adding consecutive integers. This confirms the design of the pairs.

Verification / Alternative check:
Write the final paired sequence explicitly: (3, 8), (5, 27), (8, 64), (12, 125), (17, 216). The first term in each pair follows the progression 3, 5, 8, 12, 17, where the increments are 2, 3, 4, 5. The second term in each pair follows 2^3, 3^3, 4^3, 5^3, 6^3. This double structure is very clean and exactly matches all given values, so 216 is fully consistent as the missing term.

Why Other Options Are Wrong:
361, 625, and 441 are all perfect squares (19^2, 25^2, and 21^2 respectively) but the series of second terms is clearly following perfect cubes, not squares. Also, moving from 125 to any of these numbers does not maintain the pattern of cubes of consecutive integers. Therefore none of these alternatives can be correct under the discovered rule.

Common Pitfalls:
Some candidates try to treat the entire ten term series as one single sequence, which makes the pattern look very complex. The crucial step is to recognise the pair structure and analyse first and second elements separately. Another mistake is to spot only the cubes and ignore the subtle but simple pattern in the first elements, which can lead to uncertainty about whether the cube pattern truly extends to 6^3.

Final Answer:
The missing number in the series is the next cube, which is 216.