Pattern guide
Arithmetic-Geometric Hybrid
Each term is the previous term multiplied by a fixed number, then shifted by a constant.
A Hybrid sequence combines multiplication and addition into one rule: next = previous × a + b. The differences between terms are not constant, and the ratios are not constant either — which is why it does not look like a simple pattern at first. The key is to find the two-part rule.
How to recognise it
- 1Calculate differences — they are not constant (so not arithmetic).
- 2Calculate ratios — they are not constant (so not pure geometric).
- 3Try: does each term = previous × 2 + something? Check if the 'something' is always the same.
- 4Common multipliers: ×2, ×3. Common additions: +1, -2, +5.
- 5If you know the multiplier, rearrange: extra = current - (previous × multiplier). If that extra is always the same, you found the rule.
Worked examples
Example 1
2
5
11
23
47
?
Rule: Each term = previous × 2 + 1
2 → 5(2 × 2 + 1 = 5)
5 → 11(5 × 2 + 1 = 11)
11 → 23(11 × 2 + 1 = 23)
23 → 47(23 × 2 + 1 = 47)
47 → ?(47 × 2 + 1 = 95)
Answer: 47 × 2 + 1 = 95
Example 2
1
4
13
40
121
?
Rule: Each term = previous × 3 + 1
1 → 4(1 × 3 + 1 = 4)
4 → 13(4 × 3 + 1 = 13)
13 → 40(13 × 3 + 1 = 40)
40 → 121(40 × 3 + 1 = 121)
121 → ?(121 × 3 + 1 = 364)
Answer: 121 × 3 + 1 = 364
Example 3
6
9
15
27
51
?
Rule: Each term = previous × 2 - 3
6 → 9(6 × 2 - 3 = 9)
9 → 15(9 × 2 - 3 = 15)
15 → 27(15 × 2 - 3 = 27)
27 → 51(27 × 2 - 3 = 51)
51 → ?(51 × 2 - 3 = 99)
Answer: 51 × 2 - 3 = 99
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