A number series is given with one term missing. Study the pattern in 5, 11, 24, 51, 106, ? and choose the correct alternative that completes the series.

Introduction / Context:
This number series question is more challenging and is suitable for higher-level competitive exams. The sequence is 5, 11, 24, 51, 106, ?, and we must determine the next term. The growth rate is faster than simple addition but not strictly geometric, suggesting that the differences between terms themselves follow a pattern. Understanding such second-level patterns is crucial for advanced number series problems.

Given Data / Assumptions:
Series: 5, 11, 24, 51, 106, ?
Exactly one term is missing at the end of the series.
The sequence increases rapidly, hinting at multiplication or doubling within the pattern of differences.
We assume a single consistent rule connects each term to the next.

Concept / Approach:
First, we calculate the differences between consecutive terms to see whether those differences form a recognisable sequence. If those differences themselves roughly double or follow a simple recurrence, we can extend that pattern to calculate the next difference and therefore the next term. This is a typical approach for complex series where the original numbers grow quickly.

Step-by-Step Solution:
Compute first-level differences. 11 - 5 = 6. 24 - 11 = 13. 51 - 24 = 27. 106 - 51 = 55. Now observe the pattern in the differences: 6, 13, 27, 55. Each difference is nearly double the previous one plus 1. 6 * 2 + 1 = 13, 13 * 2 + 1 = 27, 27 * 2 + 1 = 55. Applying the same rule: next difference = 55 * 2 + 1 = 111. Add this to the last term: 106 + 111 = 217. Thus, the missing term is 217.

Verification / Alternative check:
We summarise the pattern as: difference(n) = difference(n - 1) * 2 + 1. Starting from 6, this generates 6, 13, 27, 55, 111. Adding these differences successively to the initial term 5 recreates the series: 5, 11, 24, 51, 106, 217. This structure is smooth, systematic, and typical of higher difficulty exam questions, which confirms the validity of 217 as the next term.

Why Other Options Are Wrong:
Option A: 122 gives a last difference of only 16, which does not fit the doubling plus 1 pattern for differences.
Option C: 120 gives a difference of 14, again inconsistent with the rapidly growing differences 6, 13, 27, 55.
Option D: 153 gives a difference of 47, which does not match any simple transformation of 55 such as doubling plus 1.
Option E: 211 gives a difference of 105, still not equal to 55 * 2 + 1.

Common Pitfalls:
One pitfall is to search for a direct multiplicative relation between each pair of terms rather than looking at the pattern in the differences. Another mistake is to assume a fixed difference or ratio when the series clearly grows faster than that. For tougher number series, it is vital to explore second-level patterns such as how the differences themselves change from step to step.

Final Answer:
The next term, based on the pattern where each difference is double the previous difference plus one, is 217, so the correct option is 217.