Series Completion

What you'll learn

• Identify arithmetic and geometric patterns in number sequences
• Spot mixed-rule series that combine two operations
• Decode letter series using alphabet positions
• Find missing terms and the next term in any series

Key concepts

What is a Series?

A series is an ordered list of numbers or letters where each term follows a consistent rule. Your task is to identify the rule and find the missing or next term.

Type 1 — Arithmetic Series

Each term is obtained by adding or subtracting a fixed number (the common difference).

Worked Example: 4, 9, 14, 19, 24, ?

• Differences: 5, 5, 5, 5 → common difference = +5
• Next term: 24 + 5 = 29

Tip: Always subtract consecutive terms to find the difference. If all differences are equal, it is an arithmetic series.

Type 2 — Geometric Series

Each term is obtained by multiplying or dividing by a fixed number (the common ratio).

Worked Example: 3, 12, 48, 192, ?

• Ratios: 12÷3=4, 48÷12=4, 192÷48=4 → common ratio = ×4
• Next term: 192 × 4 = 768

Tip: Divide each term by the one before it. If all ratios are equal, it is a geometric series.

Type 3 — Mixed / Two-step Series

The rule alternates or uses two operations together.

Pattern A — Alternating operations: 2, 3, 6, 7, 14, 15, ?

• +1, ×2, +1, ×2, +1 → next: ×2 → 15 × 2 = 30

Pattern B — Difference of differences (second-order): 1, 2, 4, 7, 11, 16, ?

• Differences: 1, 2, 3, 4, 5 → second difference = +1 each time
• Next difference: 6 → 16 + 6 = 22

Pattern C — Squares or cubes: 1, 4, 9, 16, 25, ?

• 1², 2², 3², 4², 5² → next: 6² = 36

1, 8, 27, 64, 125, ?

• 1³, 2³, 3³, 4³, 5³ → next: 6³ = 216

Type 4 — Letter Series

Letters follow patterns based on their alphabet position.

Basic forward/backward steps: A, C, E, G, ?

• Skip one letter each time (+2) → next: I

Z, X, V, T, ?

• Subtract 2 each time → next: R

Worked Example — Position-based: B, E, H, K, ?

• B(2), E(5), H(8), K(11) → +3 each time → 11+3=14=N

Worked Example — Two-group pattern: A, B, D, E, G, H, J, ?

• Pairs: (A,B) skip (C) (D,E) skip (F) (G,H) skip (I) → next pair starts: J, K... next missing letter = K

Important: For letter series, always convert letters to numbers first. This makes the pattern much easier to spot.

Finding a Missing Term (not always the last)

3, 7, ?, 15, 19

• Differences should all be +4 → 7 + 4 = 11

A, D, ?, J, M

• Positions: 1, 4, ?, 10, 13 → difference = +3 → 4+3 = 7 = G

Quick Reference: How to Identify the Type

Quick check

1. Find the next term: 5, 11, 17, 23, ?
2. Find the next term: 4, 12, 36, 108, ?
3. Find the missing term: 2, 6, 12, ?, 30, 42
4. Find the next letter: D, G, J, M, ?
5. Find the next term: 1, 1, 2, 3, 5, 8, ?

Open the Practice tab for graded questions on Number and Letter Series.