An alternating sequence uses a repeating cycle of operations between adjacent terms. The values can rise and fall, so searching only for a steadily increasing pattern can miss the structure.
Label every arrow
Consider 4, 12, 10, 30, 28. The transitions are multiply by three, subtract two, multiply by three, subtract two. Continuing the cycle gives 28 × 3 = 84.
The final subtraction does not mean the next step is another subtraction. You need to know where you are in the operation cycle.
| Transition | Operation |
|---|---|
| 4 to 12 | ×3 |
| 12 to 10 | −2 |
| 10 to 30 | ×3 |
| 30 to 28 | −2 |
| 28 to the next term | ×3 |
Keep the repeated rule constant
In a genuine candidate pattern, the same cycle should account for all supplied transitions. If you need a different exception at every step, the explanation is probably not useful.
For 20, 10, 15, 7.5, 12.5, the cycle divide by two, add five gives a next term of 6.25. Fractions do not invalidate a rule; consistency matters more than whether every result is an integer.
Alternating operations are not separate streams
In an operation cycle, each term comes from the immediately previous term. In an interleaved sequence, odd-positioned terms may form one pattern and even-positioned terms another. Both are worth checking, but they are different hypotheses.
Try this one
Continue 5, 9, 18, 22, 44. The operations alternate add four and multiply by two, so the next value is 48.
When checking your answer, work backward one step as well: 48 minus four returns 44. This will not prove the whole pattern by itself, but it can catch a calculation mistake after you have established the cycle.