In the series 3, 7, 6, 5, 9, 3, 12, 1, 15, ..., determine the next number by analysing the two interleaved subsequences.

Introduction / Context:
This number series alternates up and down: 3, 7, 6, 5, 9, 3, 12, 1, 15, ... . The task is to find the next term. Many reasoning questions use interleaved subsequences where the first, third, fifth and other odd positions follow one rule, and the second, fourth, sixth and other even positions follow another rule.

Given Data / Assumptions:
The given terms are:

• 3
• 7
• 6
• 5
• 9
• 3
• 12
• 1
• 15
• ?

We assume that there are two simple and consistent patterns, one for the numbers in odd positions and one for the numbers in even positions.

Concept / Approach:
Whenever consecutive terms appear to jump in a non uniform way, it is helpful to separate the series into odd and even positions. Here, the odd position numbers show an increasing trend, while the even position numbers show a decreasing trend. Recognizing these as two arithmetic progressions leads directly to the missing term.

Step-by-Step Solution:
Step 1: List the numbers in odd positions (1st, 3rd, 5th, 7th, 9th): 3, 6, 9, 12, 15.Step 2: Notice that this subsequence increases by 3 each time: 3, 6, 9, 12, 15, ... . This is an arithmetic progression with common difference +3.Step 3: List the numbers in even positions (2nd, 4th, 6th, 8th, and the unknown 10th): 7, 5, 3, 1, ?.Step 4: Observe that this subsequence decreases by 2 each time: 7, 5, 3, 1, ... . So the next term should be 1 - 2 = -1.Step 5: Position 10 is an even position, so we take the next value from the decreasing subsequence, which is -1.

Verification / Alternative check:
We can now rebuild the series by interleaving the two subsequences:
Odd positions: 3, 6, 9, 12, 15Even positions: 7, 5, 3, 1, -1Interleaved series: 3, 7, 6, 5, 9, 3, 12, 1, 15, -1.
This matches all the given terms and supplies a consistent next term without any contradiction, so the pattern is correct.

Why Other Options Are Wrong:
Values such as 2, 0 or 1 do not continue the even position subsequence 7, 5, 3, 1 with the same common difference of -2. Using any of them would break the arithmetic progression in the even positions, so they are not compatible with the observed pattern.

Common Pitfalls:
Students sometimes try to find a single rule linking each term directly to the previous one, which is difficult in interleaved series. The key skill is to recognise that odd and even positions may follow different rules, and to separate them before searching for patterns like arithmetic progressions.

Final Answer:
The next number in the series, following the pattern of two interleaved progressions, is -1.