A number series is given with one term missing. Choose the correct alternative from the options that will complete the series: 79, 159, 199, 219, ?

Introduction / Context:
This question involves a number series where the differences between consecutive terms form a pattern related to halving. Recognizing that the increments themselves shrink in a structured way is essential. Such patterns test the ability to observe changes in differences rather than just in the original numbers.

Given Data / Assumptions:
Given series: 79, 159, 199, 219, ?We must find the next term after 219.We assume that the differences between terms follow a simple, repeating, or steadily changing rule.

Concept / Approach:
We start by calculating the differences between consecutive terms. If these differences themselves follow a pattern, such as halving or doubling, we extend that pattern to obtain the next difference and thus the missing term. This technique is standard when the primary sequence appears irregular at first glance but the steps between terms show more regularity.

Step-by-Step Solution:
Step 1: Compute the difference 159 - 79 = 80.Step 2: Compute the difference 199 - 159 = 40.Step 3: Compute the difference 219 - 199 = 20.Step 4: The differences are 80, 40, 20, forming a sequence where each difference is half of the previous one.Step 5: Continuing this pattern, the next difference should be 10, which is half of 20.Step 6: Add this difference to the last term: 219 + 10 = 229.

Verification / Alternative check:
Write the series including the candidate term: 79, 159, 199, 219, 229. Now compute all differences: 159 - 79 = 80, 199 - 159 = 40, 219 - 199 = 20, and 229 - 219 = 10. The pattern of differences 80, 40, 20, 10 consistently halves each time, confirming that 229 is the correct next term in the sequence.

Why Other Options Are Wrong:
Option 234 would give a difference of 15 from 219, which does not match the halving pattern of 80, 40, 20, 10.Option 239 would produce a difference of 20, repeating the previous difference instead of halving it, and thereby breaking the established pattern.Option 222 would yield a difference of 3, which fits no simple relationship with the earlier differences of 80, 40, and 20.

Common Pitfalls:
Students sometimes search for a direct rule among the original numbers and overlook the more obvious pattern in the differences. Another frequent mistake is failing to compute differences correctly, especially when several values are involved in a chain. Always calculate differences systematically and check whether they follow a simple structure like halving or doubling.

Final Answer:
The missing term that continues the halving-difference pattern is 229.