In the series 6, 12, 21, x, 48, identify the value of x that continues the same number pattern between the terms.

Introduction / Context:
This is a moderately simple increasing series with one missing term. The jumps between numbers grow in a regular way, which suggests that the differences between consecutive terms form their own arithmetic pattern. Determining these differences and extending the pattern enables us to find x.

Given Data / Assumptions:

• The known terms are 6, 12, 21, x, 48.
• Only one term, x, is missing.
• A single, consistent rule is assumed to generate the progression.
• Differences between successive terms are the natural first place to look for structure.

Concept / Approach:
We compute and inspect the first level differences. If these differences themselves follow an arithmetic pattern (for example, they increase by a constant amount), then we can extend that difference pattern to find the missing value and thus recover x. This kind of two stage progression is very common in number series questions.

Step-by-Step Solution:
Step 1: Compute known differences between consecutive terms. 12 - 6 = 6. 21 - 12 = 9. Step 2: Let x be the next term and 48 be the last term. Then the remaining differences are x - 21 and 48 - x. Step 3: Look at the existing differences: 6 and 9. The increase from 6 to 9 is +3, so it is reasonable to expect the next differences to be 12 and 15, continuing this pattern: 6, 9, 12, 15. Step 4: Set x - 21 = 12, because this should be the third difference. Solving, x = 21 + 12 = 33. Step 5: Check the last difference: 48 - x = 48 - 33 = 15, which matches the expected fourth difference. Step 6: Therefore x = 33 fits perfectly.

Verification / Alternative check:
Write out the completed series with x = 33: 6, 12, 21, 33, 48. Now list the differences: 12 - 6 = 6, 21 - 12 = 9, 33 - 21 = 12, 48 - 33 = 15. The differences are 6, 9, 12, 15, an arithmetic sequence with common difference 3. This neat structure confirms that we have identified the correct missing term.

Why Other Options Are Wrong:
If x were 30, the differences would be 6, 9, 9, 18, which break the +3 pattern. If x were 42, the last difference 48 - 42 would be 6 instead of 15. Similarly, x = 45 or any other value not equal to 33 fails to create a smooth arithmetic progression of differences. Thus these alternatives do not maintain the simple rule that governs the series.

Common Pitfalls:
One common pitfall is to try only a couple of options and test them numerically, rather than systematically deriving the pattern in the differences. Another is to assume that the differences should be constant, which does not fit here. Recognizing an arithmetic series in the first level differences is the key to an efficient and reliable solution.

Final Answer:
The value of x that continues the pattern of increasing differences (6, 9, 12, 15) is 33.