Pattern guide
Alternating Operator
Two different rules take turns. Odd positions follow one rule, even positions follow another.
An Alternating Operator sequence applies two operations in rotation. Instead of one pattern running through all terms, you have two patterns interleaved: operation A on odd steps, operation B on even steps. The trick is recognising the alternation.
How to recognise it
- 1If the differences look inconsistent, check whether they alternate between two types.
- 2Look at every other step: does step 1→2 and step 3→4 do the same thing? Does step 2→3 and step 4→5 do the same thing?
- 3Common pairs: ×2 then -3, +5 then ×2, +odd then ×even.
- 4A common giveaway: the sequence alternates between going up by a lot and coming down a little (or vice versa).
- 5List the operations explicitly: ×2, -3, ×2, -3... and see if a two-step cycle emerges.
Worked examples
Example 1
4
8
5
10
6
12
7
?
Rule: Odd steps: +1. Even steps: ×2
4 → 8(×2 (even step))
8 → 5(+(-3)? No — let us re-read the pattern)
4 → 5(odd positions: 4, 5, 6, 7 — each +1)
8 → 10(even positions: 8, 10, 12 — each +2)
7 → ?(next is an even position: 12 + 2 = 14)
Answer: Next even-position term: 12 + 2 = 14
Example 2
10
20
17
34
31
62
?
Rule: Odd→even: ×2. Even→odd: -3
10 → 20(×2)
20 → 17(-3)
17 → 34(×2)
34 → 31(-3)
31 → 62(×2)
62 → ?(-3)
Answer: 62 - 3 = 59
Example 3
3
9
12
36
39
117
?
Rule: Odd→even: ×3. Even→odd: +3
3 → 9(×3)
9 → 12(+3)
12 → 36(×3)
36 → 39(+3)
39 → 117(×3)
117 → ?(+3)
Answer: 117 + 3 = 120
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