A series is given with one term missing. Choose the correct alternative from the given ones that will complete the series: 14, 22, 33, 47, 64, ?

Introduction / Context:
This number series increases with non constant steps, but the increases themselves follow a simple pattern. In particular, the differences between consecutive terms form an arithmetic sequence. Identifying and extending that sequence of differences allows you to determine the missing next term efficiently.

Given Data / Assumptions:

• Series: 14, 22, 33, 47, 64, ?
• Exactly one next term is missing.
• The term to term increments appear to grow gradually.

Concept / Approach:
Compute first differences between consecutive terms. If these differences form their own arithmetic progression, we can extend that progression to find the next difference. Adding this new difference to the last term of the original series yields the missing term. This method leverages the idea of a sequence built from another simpler sequence of step sizes.

Step-by-Step Solution:
Step 1: Compute the differences between consecutive terms. 22 - 14 = 8. 33 - 22 = 11. 47 - 33 = 14. 64 - 47 = 17. Step 2: Collect these differences: 8, 11, 14, 17. Step 3: Note that the differences form an arithmetic sequence, increasing by 3 each time: 8, 11, 14, 17. Step 4: The next difference should therefore be 17 + 3 = 20. Step 5: Add this next difference to the last known term of the series: 64 + 20 = 84.

Verification / Alternative check:
Extend the series with the computed number: 14, 22, 33, 47, 64, 84. Recalculate the differences: 8, 11, 14, 17, 20. These differences clearly form an arithmetic progression with common difference 3. This confirms that the underlying structure of the series is consistent and that 84 is the correct missing term.

Why Other Options Are Wrong:
If we choose 81, 92, or 94, the last difference will not be 20 and will break the +3 pattern in the difference sequence. For instance, if the next term were 81, the final difference would be 81 - 64 = 17, repeating rather than increasing, which contradicts the observed progression 8, 11, 14, 17. Therefore those alternatives are incompatible with the established rule.

Common Pitfalls:
Some candidates attempt to guess the next term by approximate growth without fully checking the differences. Others may look for multiplicative patterns among the main terms, which are not present here. The most reliable approach is to calculate all consecutive differences and examine them carefully for simple sequences like constant increments or decrements.

Final Answer:
The number that correctly completes the series is 84.