In the following question, select the missing number from the given series: 2, 4, 13, 41, 106, ?

Introduction / Context:
This series grows quite rapidly, and the step sizes between consecutive terms increase in a non linear way. Problems like this often hide a pattern in the differences between terms, where those differences follow a rule related to squares or cubes. In this question, the differences are based on cubes of natural numbers plus one, which is a nice example of using an auxiliary sequence to control the growth of the main series.

Given Data / Assumptions:

• Series: 2, 4, 13, 41, 106, ?
• Exactly one next term is missing.
• The differences between terms increase and are unlikely to be constant.

Concept / Approach:
Calculate the differences between consecutive terms and then see if these differences themselves form a recognisable pattern. Here, each difference turns out to be one more than a perfect cube. Once we recognise that the differences are 1^3 + 1, 2^3 + 1, 3^3 + 1, and so on, we can immediately compute the next difference and hence the missing term in the series.

Step-by-Step Solution:
Step 1: Compute the first differences. 4 - 2 = 2. 13 - 4 = 9. 41 - 13 = 28. 106 - 41 = 65. Step 2: Observe these differences: 2, 9, 28, 65. Step 3: Express each difference in terms of cubes. 2 = 1^3 + 1. 9 = 2^3 + 1. 28 = 3^3 + 1. 65 = 4^3 + 1. Step 4: The pattern is difference(n) = n^3 + 1 for n = 1, 2, 3, 4, ... Step 5: The next difference should be 5^3 + 1 = 125 + 1 = 126. Step 6: Add this to the last known term 106: 106 + 126 = 232.

Verification / Alternative check:
Extend the series with the computed term: 2, 4, 13, 41, 106, 232. Now recompute the differences: 2, 9, 28, 65, 126. These match exactly 1^3 + 1, 2^3 + 1, 3^3 + 1, 4^3 + 1, 5^3 + 1. The pattern is therefore perfectly consistent, confirming that 232 is the correct missing term and that the underlying structure of the series is based on cubes plus one.

Why Other Options Are Wrong:
Values 172, 191, and 219 do not produce a final difference that equals 126. For example, 191 - 106 = 85, which is not equal to 5^3 + 1. Using any of these alternatives would break the clean sequence of differences being exact cube plus one values, making the series inconsistent with the discovered rule.

Common Pitfalls:
It is tempting to try proportional or multiplicative relationships directly on the main terms, which quickly becomes messy here. Another common error is to look for square based differences rather than cube based ones, since squares are more familiar. The key is to systematically test small powers and see which ones fit, rather than guessing. Recognising 2, 9, 28, 65 as cubes plus one is the decisive step in solving this problem.

Final Answer:
The next number in the series is 232.