In the following number series, one term is missing. Using the observed pattern of multiplication, find the missing number: 19, 38, ?, 228, 684, 1368.

Introduction / Context:
This question features a number series where each term is generated by multiplying the preceding term by a changing factor. Such alternating multiplication patterns, especially involving factors like 2 and 3, are very common in reasoning tests and need to be recognized quickly.

Given Data / Assumptions:
The series provided is:
19, 38, ?, 228, 684, 1368
Our assumption is that a multiplicative rule, possibly alternating between two factors, governs the transitions between terms. We need to identify this rule and use it to compute the missing third term.

Concept / Approach:
A natural first step is to check for simple multipliers. From 19 to 38, we see the clear factor of 2. Next, from 228 to 684 and from 684 to 1368, we can check their ratios and see if there is a pattern like multiplying by 2, then by 3, and so on. Once the pattern is clear at multiple positions, we can safely apply it where a term is missing.

Step-by-Step Solution:
Step 1: Check the ratio between known consecutive terms at the end of the series.228 to 684: 684 / 228 = 3, so this step multiplies by 3.684 to 1368: 1368 / 684 = 2, so this step multiplies by 2.Step 2: Observe that the series might alternate between multiplying by 2 and multiplying by 3.Step 3: Check the beginning of the series with this hypothesis.19 to 38 is multiplication by 2.Therefore, from 38 to the missing term we should multiply by 3.Step 4: Multiply 38 by 3 to find the missing term.38 * 3 = 114.Step 5: Confirm by continuing the pattern.From 114 to 228, we should multiply by 2, and indeed 114 * 2 = 228, which matches the series.

Verification / Alternative check:
Write the full pattern explicitly: start with 19, then multiply by 2 to get 38, multiply by 3 to get 114, multiply by 2 to get 228, multiply by 3 to get 684, and finally multiply by 2 to get 1368. This alternating multiplication by 2 and 3 perfectly fits all known terms and explains the entire series smoothly, which confirms that 114 is the correct missing value.

Why Other Options Are Wrong:
Numbers like 108, 113, 138, or 120 do not maintain the strict pattern of alternating factors. For example, if we chose 108, the ratio from 38 to 108 would not be 3, and the ratio from 108 to 228 would not be 2. Thus, the regular structure of the series would be broken, making these options invalid.

Common Pitfalls:
Students sometimes try to force an arithmetic pattern on such series or miscalculate one of the ratios, which sends them in the wrong direction. It is critical to compute ratios carefully and to check consistency across multiple segments of the series before confirming a rule.

Final Answer:
The missing term that preserves the alternating multiplication pattern is 114.