In the number series 19, 25, 32, 38, 45, 51, ?, which number should come next to maintain the existing pattern?

Introduction / Context:
This series increases steadily with moderately sized steps. The key idea is to inspect the differences between consecutive terms and see whether these differences themselves follow a simple repeating pattern. Such alternating-difference sequences are very common in competitive exams.

Given Data / Assumptions:
- The series is: 19, 25, 32, 38, 45, 51, ?- All terms are positive integers increasing by small amounts.- The differences appear to alternate between two nearby values.

Concept / Approach:
We calculate the differences between consecutive terms and look for repetition or alternation. If the differences alternate between two constants, we simply continue that alternation to find the next difference and thus the next term. This approach quickly exposes the underlying rule in such series questions.

Step-by-Step Solution:
- Compute consecutive differences: 25 - 19 = 6 32 - 25 = 7 38 - 32 = 6 45 - 38 = 7 51 - 45 = 6- So the differences form the pattern: 6, 7, 6, 7, 6.- The next difference should continue this pattern, so it must be 7.- Add this difference to the last term: 51 + 7 = 58.- Therefore, the missing term is 58.

Verification / Alternative check:
- Rebuild the series using alternating differences 6 and 7: 19 + 6 = 25, 25 + 7 = 32, 32 + 6 = 38, 38 + 7 = 45, 45 + 6 = 51, 51 + 7 = 58.- Every term matches the original series, and the newly obtained term fits perfectly at the end.

Why Other Options Are Wrong:
- 59, 62, and 64 do not correspond to adding 7 to 51.- Adopting any of these options would break the neat alternating 6, 7 difference pattern.- Only 58 preserves the exact alternation structure and remains consistent with all previous terms.

Common Pitfalls:
Some candidates overlook the alternating nature of the differences and look for a single constant step or more complicated formula. Others miscalculate one of the differences and fail to see the 6, 7, 6, 7 pattern. Carefully computing and writing out each difference can quickly reveal such alternating structures.

Final Answer:
The number that should come next in the series is 58, so the correct option is 58.