Series & Patterns

Core Idea

Find the rule that connects consecutive terms, then apply it to the next term. Always check differences first, then ratios, then differences-of-differences.

Types of Number Series

• Arithmetic (constant difference): 5, 9, 13, 17... (+4 each time)
• Geometric (constant ratio): 3, 6, 12, 24... (×2 each time)
• Changing difference (arithmetic of differences): 2, 5, 10, 17, 26... (differences 3,5,7,9 — add consecutive odd numbers)
• Squares/cubes: 1, 4, 9, 16, 25... (n²) or 1, 8, 27, 64... (n³)
• Alternating operations: 2, 6, 5, 9, 8, 12... (+4, -1, +4, -1...)
• Two-step (multiply then add/subtract): 3, 7, 15, 31... (×2+1 each time)

Types of Letter Series

• Track position value: A=1, B=2, C=3...Z=26
• Skip pattern: B, D, F, H... (skip 1 letter, +2 positions)
• Reverse skip: Z, X, V, T... (skip 1 letter backward, -2 positions)
• Double letter shift: AZ, BY, CX... (first letter +1, second letter -1)

Mixed Series

• Two interleaved series combined into one sequence — separate into odd/even position sub-series first
• Example: A, 1, C, 3, E, 5... → letters at odd positions (A,C,E... skip 1), numbers at even positions (1,3,5... +2) → next = G

Worked Examples

1. 3, 6, 12, 24, ? → ×2 pattern → 48
2. Z, X, V, T, ? → skip 1 letter backward → R
3. 7, 10, 16, 25, 37, ? → differences 3,6,9,12 → next diff 15 → 52
4. 2, 3, 5, 8, 13, ? (each term = sum of previous two) → 21
5. AZ, CX, EV, ? → first letter +2 (A,C,E,G), second letter -2 (Z,X,V,T) → GT

Drill

1. 5, 11, 23, 47, ? — 95 (×2+1)
2. 100, 90, 81, 73, ? — 66 (differences -10,-9,-8,-7)
3. B, E, H, K, ? — N (+3 each)
4. 1, 4, 9, 16, 25, ? — 36 (squares)
5. 2, 6, 12, 20, 30, ? — 42 (n×(n+1))
6. A, C, F, J, ? — O (+2,+3,+4,+5 positions)

Practice tracker

YYYY-MM-DD

• Attempted/Correct:
• Mistake pattern: