Number and Alphabet Series — Study Notes (SOF IMO)

Overview

Number and alphabet series problems are a staple of the SOF IMO Logical Reasoning section. These questions test your ability to identify patterns and relationships between consecutive terms in a sequence of numbers, letters, or a combination of both. Typically, you'll be given 4–6 terms and asked to find the next term or a missing element in the middle.

Success in this topic requires two skills: recognising common pattern types quickly and calculating accurately under time pressure. Most series follow arithmetic or geometric progressions, prime/square/cube sequences, or simple alphabet shifts. The trick is to spot which pattern applies in the first 10–15 seconds, then verify your answer by checking if the rule holds for all given terms.

This topic carries 2–4 questions in the exam. Master the standard patterns and you'll score full marks here with minimal time investment, leaving more time for tougher geometry or reasoning questions.

Key Concepts

• **Arithmetic progression**: Each term increases or decreases by a constant difference (e.g., 3, 7, 11, 15 → difference +4).
• **Geometric progression**: Each term is multiplied or divided by a constant ratio (e.g., 2, 6, 18, 54 → ratio ×3).
• **Square/cube sequences**: Terms follow n², n³ or related patterns (e.g., 1, 4, 9, 16, 25 are squares of 1, 2, 3, 4, 5).
• **Prime number series**: Sequence of consecutive primes (2, 3, 5, 7, 11, 13...).
• **Alternating patterns**: Two interleaved sub-series, each with its own rule (e.g., odd positions increase by 3, even positions double).
• **Alphabet series**: Letters advance by fixed positions (A+3=D), or follow patterns like vowels, consonants, or reverse alphabet coding (A↔Z, B↔Y...).
• **Mixed alphanumeric**: Separate the letter and number components, identify each pattern independently, then combine.
• **Two-step or nested patterns**: The differences themselves form a series (e.g., differences are 2, 4, 6, 8...).

Formulas / Key Facts

• **Arithmetic series nth term**: aₙ = a₁ + (n−1)d, where d is the common difference.
• **Geometric series nth term**: aₙ = a₁ × r⁽ⁿ⁻¹⁾, where r is the common ratio.
• **Square of n**: 1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100.
• **Cube of n**: 1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 6³=216, 7³=343, 8³=512.
• **First 15 primes**: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
• **Alphabet positions**: A=1, B=2, C=3... Z=26. Remember this for coding problems.
• **Reverse alphabet**: A↔Z (1↔26), B↔Y (2↔25), C↔X (3↔24)... Position + reverse position always = 27.
• **Fibonacci-like sequences**: Each term is the sum of the previous two (e.g., 1, 1, 2, 3, 5, 8, 13...).

Worked Examples

**Example 1: Arithmetic progression** Series: 5, 9, 13, 17, ? *Solution*: Calculate differences: 9−5=4, 13−9=4, 17−13=4. Common difference d=4. Next term = 17+4 = **21**.

**Example 2: Geometric progression** Series: 3, 12, 48, 192, ? *Solution*: Calculate ratios: 12÷3=4, 48÷12=4, 192÷48=4. Common ratio r=4. Next term = 192×4 = **768**.

**Example 3: Square sequence** Series: 4, 9, 16, 25, ? *Solution*: Recognise 2²=4, 3²=9, 4²=16, 5²=25. Next is 6²= **36**.

**Example 4: Alternating pattern** Series: 2, 5, 4, 10, 8, 20, ? *Solution*: Separate odd and even positions. Odd: 2, 4, 8 (each ×2). Even: 5, 10, 20 (each ×2). Next is odd position, so 8×2= **16**.

**Example 5: Alphabet series** Series: C, F, I, L, ? *Solution*: Convert to positions: C=3, F=6, I=9, L=12. Arithmetic with d=3. Next = 12+3=15, which is **O**.

**Example 6: Mixed alphanumeric** Series: A2, C6, E12, G20, ? *Solution*: Letters: A, C, E, G (skip one letter each, so next is I). Numbers: 2, 6, 12, 20. Differences: 4, 6, 8 → next difference 10, so 20+10=30. Answer: **I30**.

**Example 7: Prime series** Series: 7, 11, 13, 17, ? *Solution*: These are consecutive primes after 5. Next prime is **19**.

**Example 8: Difference series (nested pattern)** Series: 3, 5, 9, 17, 33, ? *Solution*: Differences: 2, 4, 8, 16 (each doubles). Next difference = 32, so 33+32 = **65**.

Common Mistakes

• **Assuming single pattern when alternating exists** → Always check if odd and even positions follow separate rules. If a single arithmetic/geometric rule fails, split the series.
• **Forgetting zero in position counting** → When converting letters to numbers, A=1 not 0. Students often miscount by one.
• **Miscalculating squares and cubes** → 7²=49 not 48; 4³=64 not 54. Memorise up to 10² and 8³ to avoid arithmetic errors under pressure.
• **Stopping at first-order differences** → If first differences don't form a pattern, check second differences (difference of differences). Many IMO series are quadratic (second difference constant).
• **Ignoring modulo wraparound in alphabet** → If adding positions exceeds 26, wrap around: Z+2 = B (26+2=28, 28−26=2). Don't write "28" as the answer.
• **Misreading mixed series** → In A1B2C3, separate letters (A,B,C) from numbers (1,2,3) before analysing. Students often try to find one combined rule and waste time.

Quick Reference

• Arithmetic series: constant difference between terms → add/subtract same number each time.
• Geometric series: constant ratio → multiply/divide by same number each time.
• Alternating: split odd/even positions → solve each sub-series independently.
• Squares/cubes: if terms grow fast, test n², n³ or n²+n patterns immediately.
• Alphabet: convert to numbers (A=1...Z=26), find numeric pattern, convert back.
• Primes: memorise first 15 primes for instant recognition.
• Nested patterns: if first differences irregular, check if differences themselves form a series.