2.3 Number & Letter Sequences

Number & Letter Sequences

Core Competency: Reasoning Skills — Inductive Logic & Pattern Progression
Official sample: The CBSA publishes one number-sequence Reasoning Skills item with five answer options. Letter and mixed-sequence examples below are optional practice, not a disclosed live blueprint.

In border security analytics, operational data arrives in sequential streams: travel frequency logs, commercial convoy crossing schedules, flight arrival time intervals, and automated license plate reader timestamps. Intelligence analysts and frontline officers must detect anomalous patterns or extrapolate expected baseline activity from partial sequences. Sequence exercises practise induction: infer a simple rule from the stated terms and test the next term. The live mix of sequence types is not published.

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The Anatomy of Sequence Questions on the OTEE

A sequence practice item may present a string of numbers, letters, or alphanumeric pairs with one or two missing values indicated by brackets or a question mark:

• Numerical Progression: 3, 8, 15, 24, 35, [ ? ]
• Interleaved Sequence: 4, 19, 7, 16, 10, 13, [ ? ]
• Alphanumeric Series: C04, F09, I16, L25, [ ? ]

Candidates use the supplied options to select the term matching the intended rule. The published CBSA example has five options; our practice items may have four.

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Primary Numerical Progression Models

The published OTEE sequence sample illustrates one pattern. The five structures below are practice methods, not an exhaustive official taxonomy.

1. Arithmetic Progressions (First-Order & Second-Order)

• Constant Common Difference (First-Order): A fixed value $d$ is added or subtracted between consecutive terms. $$\text{Example: } 7, 13, 19, 25, 31, \dots \quad (d = +6)$$
• Accelerating Common Difference (Second-Order): The difference between terms increases or decreases by a constant increment $(\Delta_2 = \text{constant})$.

  Terms:          2       5      10      17      26      [ 37 ]
  1st Diff (Δ1):     +3      +5      +7      +9     [ +11 ]
  2nd Diff (Δ2):         +2      +2      +2      +2
  

  If first differences are not equal, always calculate the differences between the differences.

2. Geometric & Ratio Progressions

Each term is multiplied by a constant ratio $r$ (where $r$ can be positive, negative, or fractional):

• Doubling Series: $3, 6, 12, 24, 48, [96] \quad (r = 2)$
• Alternating Signs: $5, -15, 45, -135, [405] \quad (r = -3)$
• Fractional Halving: $160, 80, 40, 20, [10] \quad (r = 0.5)$

3. Exponent & Power Series ($n^2, n^3, n^2 \pm k$)

Squares and cubes are useful patterns to practise:

• Perfect Squares: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$
• Adjusted Squares ($n^2 - 1$): $0, 3, 8, 15, 24, 35, [48]$
• Adjusted Squares ($n^2 + n$): $2, 6, 12, 20, 30, 42, [56]$
• Perfect Cubes ($n^3$): $1, 8, 27, 64, 125, [216]$

4. Recursive & Accumulator Series (Fibonacci-Style)

In a recursive series, each new term is the sum (or product) of the two preceding terms:

$$x_n = x_{n-1} + x_{n-2}$$ $$\text{Example: } 2, 3, 5, 8, 13, 21, [34]$$

Variations may add the previous two terms plus a constant (e.g., $x_n = x_{n-1} + x_{n-2} + 1$).

5. Interleaved (Alternating / Two-Track) Sequences

When a sequence alternates high and low, test whether two interleaved series fit the terms:

• Odd Positions (1st, 3rd, 5th, 7th): Governed by Rule A.
• Even Positions (2nd, 4th, 6th, 8th): Governed by Rule B.

Sequence:       5,   28,   9,   24,   13,   20,   17,   [ ? ]
Odd Indices:    5 -------> 9 -------> 13 -------> 17        (+4 each step)
Even Indices:        28 -------> 24 -------> 20 -------> [ 16 ] (-4 each step)

The target term is at Position 8 (an even index), so the answer is $20 - 4 = 16$.

Pro Tip: If a sequence goes UP, then DOWN, then UP, then DOWN, split it into odd and even positions immediately!

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Alphabetical Sequences & Caesar Ciphers

For optional letter-sequence practice, map letters to positions in the 26-letter English alphabet:

Alphabet Progression Variations

• Constant Step Skip: C (3), F (6), I (9), L (12), [ O (15) ] ($+3$ letters).
• Expanding Step Skip: B (2), D (4), G (7), K (11), [ P (16) ] (Steps: $+2, +3, +4, +5$).
• Circular / Wrap-Around Arithmetic: If a pattern adds $+5$ starting at $W (23)$: $23 + 5 = 28$. Because the alphabet has 26 letters, subtract 26: $28 - 26 = 2$, yielding B.
• Alphanumeric Hybrids: Each element combines a letter and a number governed by linked rules. In B04, D08, F16, H32, letters advance by $+2$ (B, D, F, H -> J), while numbers double ($4, 8, 16, 32 \to 64$), giving J64.

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A Four-Step Sequence Diagnostic Heuristic

For a sequence practice item, try this four-step checklist:

[Step 1: Inspect direction]
Does the sequence rise, fall, or alternate?
  -> Test simple differences and ratios first.
  -> If alternating, also test separate odd/even tracks; a negative ratio can alternate without interleaving.

[Step 2: Calculate First Differences (Δ1)]
Compute (x_{k+1} - x_k) for all adjacent pairs.
  -> Constant difference? -> Arithmetic Series (Done).
  -> Constant ratio? -> Geometric Series (Done).

[Step 3: Test Higher-Order Relationships]
If Δ1 changes systematically:
  -> Calculate Second Differences (Δ2). Is Δ2 constant? -> Quadratic/2nd-Order Series.
  -> Compare terms to squares (n^2) or cubes (n^3).
  -> Check if term equals sum of prior two terms (Fibonacci).

[Step 4: Interleaved Split]
Split series into Odd Terms [1, 3, 5, 7] and Even Terms [2, 4, 6, 8].
Analyze each sub-series independently.

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Master Reference Table of Common Patterns