In the number series 5, 14, 42, 107, 233, ?, find the missing term by examining the pattern in the successive differences.

Introduction / Context:
This number series increases quite rapidly: 5, 14, 42, 107, 233, ?. The jumps between terms are not uniform, so this is not a simple arithmetic or geometric progression. Instead, the pattern is hidden in the differences between consecutive terms, which themselves turn out to be closely related to cubes of integers.

Given Data / Assumptions:
The known terms are:

• 5
• 14
• 42
• 107
• 233
• ?

We assume the series is generated by adding certain special numbers to each term to get the next, and that these special numbers follow a clear mathematical pattern.

Concept / Approach:
When numbers grow faster than a linear progression, but not by a fixed multiplicative factor, it is often very effective to compute the first differences and then see whether those differences align with values based on squares, cubes or other standard sequences. In this question, the differences between terms are one more than perfect cubes of consecutive integers starting from 2.

Step-by-Step Solution:
Step 1: Compute the differences between consecutive terms.14 - 5 = 9.42 - 14 = 28.107 - 42 = 65.233 - 107 = 126.Step 2: Now examine these differences: 9, 28, 65, 126.Step 3: Express each difference in terms of cubes: 2^3 + 1 = 9, 3^3 + 1 = 28, 4^3 + 1 = 65, 5^3 + 1 = 126.Step 4: This shows a clear pattern: the nth difference is n^3 + 1, starting from n = 2.Step 5: The next difference should therefore be 6^3 + 1 = 216 + 1 = 217.Step 6: Add this to the last known term: 233 + 217 = 450.

Verification / Alternative check:
We can restate the rule as: starting from 5, add 2^3 + 1, then 3^3 + 1, then 4^3 + 1, then 5^3 + 1, and so on. That gives 5 + 9 = 14, 14 + 28 = 42, 42 + 65 = 107, 107 + 126 = 233. Continuing, 233 + 217 = 450. This reconstruction is consistent with all known terms and confirms 450 as the unique logical continuation.

Why Other Options Are Wrong:
Values 316, 384 and 479 are not equal to 233 + 217. If any of them were chosen, the difference from 233 would not equal 6^3 + 1, and the difference sequence would not match 2^3 + 1, 3^3 + 1, 4^3 + 1, 5^3 + 1, 6^3 + 1. Thus they break the discovered cube based pattern.

Common Pitfalls:
Many students first attempt to find a direct multiplication rule, such as multiplying by 2 or by some fraction, which does not work here. Others stop after computing first differences and fail to notice that the differences themselves follow a familiar pattern based on cubes. It is important to check whether seemingly irregular differences can be expressed in terms of standard sequences like n^2, n^3 or n^3 + 1.

Final Answer:
The missing term in the series is 450.