Introduction / Context:
Many sequences recycle through blocks where a particular “anchor” value recurs at fixed intervals. Here, the value 61 repeats at every third position, and the two numbers in between form their own decreasing progressions. Recognizing this block pattern is essential to predict the next terms.
Given Data / Assumptions:
- Sequence: 61, 57, 50, 61, 43, 36, 61
- We need positions 8 and 9 (the two terms after the last 61).
- Positions 1, 4, 7 are all 61 (an anchor every third term).
Concept / Approach:
Split the sequence into blocks of three: [61, X, Y]. Across blocks, X and Y follow their own arithmetic progressions. Determine each progression and extend them to the next block following the anchor 61.
Step-by-Step Solution:
Blocks: [61, 57, 50], [61, 43, 36], [61, ?, ?]X terms (second in each block): 57 → 43 → next = 29 (decrease by 14 each block).Y terms (third in each block): 50 → 36 → next = 22 (also decrease by 14 each block).After the anchor 61 (position 7), the next two values (positions 8 and 9) must be 29 and 22.
Verification / Alternative check:
Differences inside each block: 61→57 = −4, 57→50 = −7; second block: 61→43 = −18, 43→36 = −7. Across blocks, the X terms drop by 14 (57 to 43), and Y terms drop by 14 (50 to 36), supporting the 29 and 22 projection.
Why Other Options Are Wrong:
29 61 / 31 61: Misplace the recurring 61 (it should appear every third term, not immediately next).27 20 / 22 15: Do not follow the consistent −14 step for X and Y across blocks.
Common Pitfalls:
Missing the block-of-three structure and expecting a single-step AP for the entire list.
Final Answer:
29 22
Discussion & Comments