A number series is given with one term missing. Choose the correct alternative from the options that will complete the series: 10, 29, 66, 127, ?

Introduction / Context:
This question involves a number series where the differences between consecutive terms are not constant but increase according to a secondary pattern. To solve such problems, we often need to examine not only the first differences but also the second differences. This tests deeper numerical reasoning skills beyond simple arithmetic sequences.

Given Data / Assumptions:
Given series: 10, 29, 66, 127, ?The fifth term is missing.We assume a consistent and simple pattern exists in the first and possibly second differences.

Concept / Approach:
The approach is to compute the differences between consecutive terms and then check whether those differences themselves follow an arithmetic pattern. If the first differences increase by a constant amount, we have constant second differences, which is a hallmark of quadratic-type number patterns often used in reasoning questions. Once we identify the second difference, we can predict the next first difference and find the missing term.

Step-by-Step Solution:
Step 1: Compute first differences: 29 - 10 = 19.Step 2: Next difference: 66 - 29 = 37.Step 3: Next difference: 127 - 66 = 61.Step 4: Now compute second differences: 37 - 19 = 18 and 61 - 37 = 24.Step 5: The second differences, 18 and 24, increase by 6. We expect the next second difference to be 30.Step 6: Add this expected second difference to the last first difference: 61 + 30 = 91.Step 7: Add this new first difference to the last term to get the missing term: 127 + 91 = 218.

Verification / Alternative check:
Insert the candidate term to get the complete series: 10, 29, 66, 127, 218. Recalculate first differences: 19, 37, 61, 91. Now calculate second differences: 37 - 19 = 18, 61 - 37 = 24, 91 - 61 = 30. These second differences form an arithmetic sequence 18, 24, 30, with a common difference of 6, confirming that the pattern is consistent and that 218 fits perfectly.

Why Other Options Are Wrong:
Option 330 would yield first differences that do not maintain the increasing second-difference pattern of plus 6.Option 115 would make the series decrease from 127 to 115, breaking the overall increasing trend and the computed difference structure.Option 273 would produce very large differences that are inconsistent with the smooth progression of second differences established earlier.

Common Pitfalls:
Some students stop after checking for a constant first difference and, failing to find one, may guess randomly. Others may miscalculate second differences or fail to notice that they increase by a constant amount. A systematic approach of computing both first and second differences and observing their patterns is crucial for reliably solving such questions.

Final Answer:
The missing term that correctly completes the series is 218.