Number and Letter Series

Series problems are the bread and butter of logical reasoning. We’re given a sequence of numbers or letters, and we need to find the pattern and predict the next term (or find the wrong term). The good news is there are only about 10-12 pattern types that cover almost every question we’ll ever see. Once we learn to spot them, these become free marks.

The Alphabet Position Table

Before we dive into letter series, let’s get this memorized. Every letter series problem becomes easy once we see letters as numbers.

The trick is to memorize EJOTY — E(5), J(10), O(15), T(20), Y(25). Any letter’s position can be found quickly by counting from the nearest landmark.

Common Number Series Patterns

The Difference Method — Our Main Weapon

This is the systematic approach that works for 80% of number series problems.

Step 1: Write out the differences between consecutive terms (first differences).

Step 2: If the first differences don’t show a pattern, take differences of the differences (second differences).

Step 3: Keep going until we see a pattern (constant, arithmetic, geometric, etc.).

Example 1: Find the next term — 2, 5, 10, 17, 26, ?

Let’s take differences:

• 5-2=3, 10-5=5, 17-10=7, 26-17=9
• First differences: 3, 5, 7, 9 — they’re increasing by 2!

So the next first difference = 11. Answer: 26 + 11 = 37

We can also see: these are n² + 1 → 1+1=2, 4+1=5, 9+1=10, 16+1=17, 25+1=26, 36+1=37 ✓

Example 2: Find the next term — 1, 2, 6, 24, 120, ?

Let’s look at ratios instead of differences:

• 2/1=2, 6/2=3, 24/6=4, 120/24=5

The pattern is ×2, ×3, ×4, ×5… so next is ×6. Answer: 120 × 6 = 720

This is the factorial series! 1!, 2!, 3!, 4!, 5!, 6! = 720

Example 3: Find the next term — 3, 5, 9, 17, 33, ?

Differences: 2, 4, 8, 16 — these are powers of 2!

Next difference = 32. Answer: 33 + 32 = 65

Pattern: each term = previous term × 2 - 1. Check: 3×2-1=5 ✓, 5×2-1=9 ✓, 9×2-1=17 ✓

Example 4: Two interleaved series — 3, 7, 5, 14, 7, 21, ?

This looks messy until we separate odd and even positions:

• Odd positions: 3, 5, 7, ? → simple +2 pattern → next = 9
• Even positions: 7, 14, 21 → simple +7 pattern (or ×7 table)

Answer: 9

Example 5: Alternating operations — 2, 6, 4, 12, 10, 30, ?

Let’s trace the operations:

• 2 → 6 (×3), 6 → 4 (-2), 4 → 12 (×3), 12 → 10 (-2), 10 → 30 (×3)

The pattern is: ×3, -2, ×3, -2, ×3, -2…

Next: 30 - 2 = 28

Finding the Wrong Number

In these problems, one number in the series doesn’t fit the pattern. We find the pattern, identify which term breaks it, and report it.

Example: Which number is wrong? 2, 3, 5, 8, 13, 21, 34, 55, 89

This looks like Fibonacci (each = sum of previous two). Let’s check:

• 2+3=5 ✓, 3+5=8 ✓, 5+8=13 ✓, 8+13=21 ✓, 13+21=34 ✓, 21+34=55 ✓, 34+55=89 ✓

Wait, they all check out! But a real Fibonacci starting at 2, 3 would give 2, 3, 5, 8, 13, 21, 34, 55, 89. This is actually correct. Let’s try starting at the actual Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. The series given starts from 2 — so it’s a correct Fibonacci subsequence.

Typical wrong-number example: 2, 3, 6, 7, 14, 15, 30, 31, 64

Pattern: +1, ×2, +1, ×2, +1, ×2, +1, ×2

• 2+1=3 ✓, 3×2=6 ✓, 6+1=7 ✓, 7×2=14 ✓, 14+1=15 ✓, 15×2=30 ✓, 30+1=31 ✓, 31×2=62 ✗

The wrong number is 64 — it should be 62.

Letter Series Patterns

For letter series, we convert each letter to its number, find the number pattern, and convert back.

Common Letter Patterns

• Skip pattern: A, C, E, G → positions 1, 3, 5, 7 → skip 1 letter each time
• Increasing skip: A, B, D, G, K → positions 1, 2, 4, 7, 11 → differences +1, +2, +3, +4
• Reverse alphabet: Z=1, Y=2, X=3… (sometimes the pattern is based on reverse positions)
• Grouping: ABC, DEF, GHI → groups of 3 consecutive letters

Example: B, E, H, K, ?

Convert: 2, 5, 8, 11 → each +3

Next = 14 = N

Example: A, Z, C, X, E, ?

Two interleaved series:

• Odd positions: A, C, E → 1, 3, 5 → +2 (forward through alphabet)
• Even positions: Z, X → 26, 24 → -2 (backward through alphabet)

Next is an even position: 24 - 2 = 22 = V

Example: Z, W, T, Q, ?

Positions: 26, 23, 20, 17 → each -3

Next = 14 = N

Mixed Number-Letter Series

These combine numbers and letters in one sequence. The key is to separate the number pattern from the letter pattern.

Example: A2, C4, E8, G16, ?

• Letters: A, C, E, G → +2 each time → next = I
• Numbers: 2, 4, 8, 16 → ×2 each time → next = 32

Answer: I32

Shortcut Tips

1. If differences are constant → arithmetic series. Done.
2. If ratios are constant → geometric series. Done.
3. If differences form their own pattern → take second differences.
4. If terms are close to perfect squares or cubes → check n², n³, n²±small number.
5. If the series looks random → try separating odd and even positioned terms.
6. If operations alternate → track the operation between each pair of terms.
7. For letter series → immediately convert to numbers using A=1 through Z=26.

Common Exam Variations

• Find the missing term (middle of series, not just the end)
• Find the wrong term (one term doesn’t fit)
• Complete the group: 2, 5, 10 : 3, ?, 12 (same pattern applied to different starting numbers)
• Letter-number combined series
• Series based on digit manipulation: sum of digits, product of digits, reversing digits

Practice Problems

Problem 1: Find the next term: 7, 11, 20, 36, 61, ?

Problem 2: Find the wrong number: 1, 2, 4, 8, 16, 32, 65, 128

Problem 3: Find the next term: D, G, K, P, ?

Answers

Problem 1: Differences: 4, 9, 16, 25 → these are 2², 3², 4², 5². Next difference = 6² = 36. Answer: 61 + 36 = 97

Problem 2: Pattern is ×2 each time: 1, 2, 4, 8, 16, 32, 64, 128. The wrong number is 65 — it should be 64.

Problem 3: Convert: D=4, G=7, K=11, P=16. Differences: 3, 4, 5. Next difference = 6. Next position = 16 + 6 = 22 = V