In the number series 7, 10, 16, 28, ?, 100, what number should replace the question mark so that the pattern remains consistent?

Introduction / Context:
This series question checks whether you can detect a pattern that uses both multiplication and subtraction in a consistent way. Such series require you to test hypotheses about how each term is generated from the previous one. The sequence 7, 10, 16, 28, ?, 100 hides a simple linear relation once you look at the operations between consecutive terms carefully.

Given Data / Assumptions:
- The given terms are 7, 10, 16, 28, ?, 100. - There is a consistent rule connecting each term to the next. - The goal is to find the missing term that preserves this rule.

Concept / Approach:
- Try to express each term in relation to the previous one using a simple operation such as multiplication plus or minus a constant. - Check if a single relation like next = 2 * current - constant works across several steps. - Once the pattern is found and verified, apply it to compute the missing term and the final term to ensure consistency.

Step-by-Step Solution:
Step 1: Look at 7 and 10. One way to get 10 from 7 is 7 * 2 - 4 = 14 - 4 = 10. Step 2: Check if the same relation holds from 10 to 16. Compute 10 * 2 - 4 = 20 - 4 = 16. This matches the series. Step 3: From 16 to 28, apply the same rule. 16 * 2 - 4 = 32 - 4 = 28. Again, it matches. Step 4: Continue the same pattern to find the missing term. From 28, next term should be 28 * 2 - 4 = 56 - 4 = 52. Step 5: Confirm that applying the same rule from 52 gives the final term 100. 52 * 2 - 4 = 104 - 4 = 100, which matches the given last term.

Verification / Alternative check:
The discovered pattern is a_n+1 = 2 * a_n - 4 for all n. We tested it across all available transitions. Since it works from the first to the second term, from the second to the third, from the third to the fourth, and from the fifth to the sixth once we insert 52, the rule is fully consistent. Therefore 52 is the only value that keeps the series coherent.

Why Other Options Are Wrong:
Option A (34): Using 34 would give 28 * 2 - 4 = 52, not 34, so the rule would be broken at that point. Option B (40): 28 * 2 - 4 is not 40, so 40 does not satisfy the established relation. Option D (60): If we use 60, then 60 * 2 - 4 = 116, which does not match 100, so the pattern fails later. Option E (46): Similarly, 28 * 2 - 4 is not 46, and the relation is inconsistent if we choose this value.

Common Pitfalls:
- Many students try difference patterns (3, 6, 12, ...) and get stuck, because the sequence is not based on differences but on a direct relation. - Some may assume changing multipliers rather than noticing the simple repeated operation of doubling and then subtracting 4. - Forgetting to verify the rule with the last term leads to incorrect choices that happen to fit only one gap.

Final Answer:
The value that must replace the question mark to maintain the pattern is 52.