Introduction / Context:
This is a classic interleaved-series problem. The overall sequence is formed from two simpler subsequences—one using the odd positions and the other using the even positions. Identify each subsequence, then extend both to obtain the next two values.
Given Data / Assumptions:
- Main sequence: 6, 20, 8, 14, 10, 8, 12
- We must determine terms at positions 8 and 9.
- Interleaving (odd vs. even indices) is suspected.
Concept / Approach:
Split the list into two streams: odd-indexed terms and even-indexed terms. If each stream is a simple arithmetic progression, extend each in order to get the next two terms of the original sequence (which alternates between the two streams).
Step-by-Step Solution:
Odd positions (1,3,5,7): 6, 8, 10, 12 → increases by +2 each step.Even positions (2,4,6): 20, 14, 8 → decreases by −6 each step.Next even-position term (position 8) continues −6: 8 − 6 = 2.Next odd-position term (position 9) continues +2: 12 + 2 = 14.Therefore, the next two terms are 2 and 14.
Verification / Alternative check:
Reconstruction interleaving: (odd stream) 6,8,10,12,14… and (even stream) 20,14,8,2… Combine to get 6,20,8,14,10,8,12,2,14, which aligns with the rule.
Why Other Options Are Wrong:
14 10 / 14 14 / 4 12: Do not respect both the −6 trend on even positions and +2 on odd positions.2 18: Second value should be 14 (odd stream +2), not 18.
Common Pitfalls:
Looking only at overall differences rather than separating into two simple alternating progressions.
Final Answer:
2 14
Discussion & Comments