Pattern guide
Recursive Acceleration
The gap between terms keeps growing (or shrinking) by a fixed rule.
In a Recursive Acceleration sequence, you do not add the same number each time. Instead, the differences themselves follow a pattern — they double, triple, or grow by a fixed amount at each step. To solve it, you need two layers of subtraction.
How to recognise it
- 1Calculate the differences between consecutive terms.
- 2If the differences are not constant, calculate the differences of those differences.
- 3If the second-layer differences are constant (e.g. all +2) or multiply by a fixed number (e.g. all ×2), you have a Recursive Acceleration sequence.
- 4Common signatures: differences that double (4, 8, 16, 32), or differences that increase by the same amount (1, 2, 3, 4, 5).
- 5Watch for both growing and shrinking sequences — the acceleration can be downward too.
Worked examples
Example 1
3
7
15
31
63
?
Rule: Each difference doubles
3 → 7(difference: +4)
7 → 15(difference: +8 (4 × 2))
15 → 31(difference: +16 (8 × 2))
31 → 63(difference: +32 (16 × 2))
63 → ?(difference: +64 (32 × 2))
Answer: 63 + 64 = 127
Example 2
1
2
4
7
11
16
?
Rule: Each difference increases by 1
1 → 2(difference: +1)
2 → 4(difference: +2 (1 + 1))
4 → 7(difference: +3 (2 + 1))
7 → 11(difference: +4 (3 + 1))
11 → 16(difference: +5 (4 + 1))
16 → ?(difference: +6 (5 + 1))
Answer: 16 + 6 = 22
Example 3
50
48
44
38
30
?
Rule: Each difference grows by -2 (shrinking faster and faster)
50 → 48(difference: -2)
48 → 44(difference: -4 (2 more than before))
44 → 38(difference: -6 (2 more than before))
38 → 30(difference: -8 (2 more than before))
30 → ?(difference: -10 (2 more than before))
Answer: 30 - 10 = 20
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