In the number series 3, 7, 6, 5, 9, 3, 12, 1, 15, ( ... ), what number should come next to continue the pattern?

Introduction / Context:
This number series alternates between two different patterns, one for the odd positions and another for the even positions. The numbers appear in pairs, and analyzing how each pair changes reveals the rule for the sequence. Recognizing such paired or interleaved patterns is very important for solving more complex series questions in aptitude exams.

Given Data / Assumptions:

• The series is 3, 7, 6, 5, 9, 3, 12, 1, 15, ( ... ).
• We must determine the next term after 15.
• Observing the sequence in pairs may make the pattern clearer.
• We assume the same rule continues beyond the given terms.

Concept / Approach:
Group the terms into ordered pairs: (3, 7), (6, 5), (9, 3), (12, 1), (15, ?). In each pair, the first number appears to increase steadily, while the second number decreases steadily. If we can describe these first elements and second elements with simple rules, we can predict the missing second element in the last pair and thus find the next term in the overall series.

Step-by-Step Solution:
Group the terms as pairs: (3, 7), (6, 5), (9, 3), (12, 1), (15, ?). Observe the first elements of the pairs: 3, 6, 9, 12, 15. These first elements form an arithmetic progression with first term 3 and common difference 3. Now observe the second elements of the pairs: 7, 5, 3, 1. These second elements form an arithmetic progression with first term 7 and common difference -2. Following the pattern, the next first element is already given as 15, fitting 3, 6, 9, 12, 15. The next second element should be 1 - 2 = -1 to continue the sequence 7, 5, 3, 1, -1. Thus, the missing term in the overall series after 15 is -1.

Verification / Alternative Check:
Let us reconstruct the full series from the paired description. Using first elements 3, 6, 9, 12, 15 and second elements 7, 5, 3, 1, -1, the pairs become (3, 7), (6, 5), (9, 3), (12, 1), (15, -1). Writing them out in sequence yields 3, 7, 6, 5, 9, 3, 12, 1, 15, -1. This exactly reproduces the given part of the sequence and appends the logical next term. Since both subsequences (one increasing by 3 and one decreasing by 2) are consistent, the answer -1 is fully justified.

Why Other Options Are Wrong:

• Option 18: This would continue the first element pattern but 18 would be the first element of a new pair, not the second element following 15.
• Option 13: Does not fit into the arithmetic progression of second elements 7, 5, 3, 1, which decrease by 2 each time.
• Option 3: This value already appeared earlier as a second element and does not continue the decreasing pattern.
• Option 0: This would imply a difference of -1 from the previous second element 1, breaking the consistent step of -2.

Common Pitfalls:
Some students may only look at overall differences between consecutive terms and fail to notice the pair structure. This leads to confusion because the differences do not form a simple pattern if treated as a single sequence. Another common mistake is to assume that the pattern should always produce positive numbers and thus reject negative values, even when the arithmetic rules clearly lead to a negative term. In exam scenarios, always follow the discovered rule even if the next term turns out to be negative.

Final Answer:
The next term in the series is -1.