A series is given with one term missing. Choose the correct alternative from the given options that will complete the series: 9, 1331, 18, 121, 27, ?

Introduction / Context:
This is a mixed pattern series that alternates between two different kinds of numbers. One subsequence involves multiples of a base number, and the other subsequence involves powers of another base. You must identify both patterns and then apply them to find the missing term. Such questions test your ability to separate an interleaved series into simpler components.

Given Data / Assumptions:

- The series given is 9, 1331, 18, 121, 27, ?.- Terms in odd positions are 9, 18, 27.- Terms in even positions are 1331, 121 and the missing value.

Concept / Approach:
When a series appears irregular, it is often formed by two independent subseries interwoven. Here, looking at the odd-position terms and even-position terms separately helps. The odd terms 9, 18 and 27 suggest a simple multiple pattern, while the even terms 1331 and 121 seem like familiar powers of 11. Once we identify both patterns, we can determine the missing even-position term.

Step-by-Step Solution:
Step 1: Consider the odd-position terms: 9 (first), 18 (third), 27 (fifth).Step 2: Recognise that 9 = 9 * 1, 18 = 9 * 2 and 27 = 9 * 3, so the odd terms are multiples of 9 in increasing order.Step 3: Now look at the even-position terms: 1331 (second) and 121 (fourth).Step 4: Note that 1331 = 11^3 and 121 = 11^2, both powers of 11.Step 5: It is natural to continue this pattern with the next lower power 11^1 = 11.Step 6: Therefore, the missing sixth term in the series should be 11.

Verification / Alternative check:
Rewrite the series as two separate subseries. Odd positions: 9, 18, 27 form 9 * 1, 9 * 2, 9 * 3. Even positions: 1331, 121, 11 form 11^3, 11^2, 11^1. When we interleave them, the series is 9, 1331, 18, 121, 27, 11, which perfectly matches both identified patterns. This verifies that 11 is the correct missing term.

Why Other Options Are Wrong:
The options 13, 17 and 19 are prime numbers but they do not fit the descending power pattern of 11^3, 11^2, 11^1. Using any of these values would disrupt the smooth sequence of powers of 11 in the even positions, so they cannot be correct. Only 11 maintains the consistent rule across the series.

Common Pitfalls:
Many candidates try to find a single rule relating each term to the next, which becomes very messy here. The key insight is to separate odd and even positions when the overall sequence looks irregular. Once you notice both the multiples of 9 and the powers of 11, the missing term becomes easy to identify.

Final Answer:
The missing number that completes the series is 11.