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4.1 Number Series Patterns

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ADF YOU Session Job Opportunities Assessment

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1 Introduction & JOA Overview

1.1 What the JOA Is and Why the ADF Uses It 1.2 Current Exam Facts (51 Questions, 20 Minutes) 1.3 JOA vs YOU Session vs the Wider Process

2 Logistics, Access & Attempt Rules

2.1 Access, Devices & Browser Setup 2.2 Three Attempts & Six-Month Wait 2.3 After the JOA: JOR and Recruiter Contact

3 Official Tips & Test-Day Strategy

3.1 Official Top Tips & Natural Ability 3.2 Pacing, Flagging & Order of Attack 3.3 Integrity, Help Rules & Supervised Verification

4 Numerical Reasoning: Series & Matrices

4.1 Number Series Patterns 4.2 Number Matrices & Missing Values 4.3 Numerical Speed & Estimation Under Time

5 Numerical Reasoning: Word Problems

5.1 Rates, Distance & Averages 5.2 Ratios, Proportions & Percentage Reasoning 5.3 Multi-Step Numerical Reasoning

6 Verbal Reasoning

6.1 Word Meaning & Relationships 6.2 Verbal Analogies 6.3 Verbal Odd-One-Out 6.4 Verbal Logic & Multi-Statement Proofs

7 Abstract Reasoning: Sequences & Rules

7.1 Sequence & Next-Item Patterns 7.2 Rotation, Reflection & Orientation 7.3 Multi-Attribute Pattern Rules

8 Abstract Reasoning: Odd-One-Out & Complex Figures

8.1 Shape Odd-One-Out 8.2 Complex Figures & Spatial-Style Abstract Items 8.3 Abstract Speed Habits

9 Integrated Prep & Common Traps

9.1 Mixed Timed Practice 9.2 Common Traps Across All Item Types 9.3 Day-of Checklist

10 Results & Next Steps

10.1 Understanding the Job Opportunities Report 10.2 Pathway After the JOA

4.1 Number Series Patterns

Key Takeaways Explain This Topic AI

JOA number series items test pattern spotting under time, not algebra proofs or school syllabus theorems.

Start with first differences; if those form a new pattern, check second differences or a multiply-then-add rule.

Alternating series hide two interleaved rules — separate odd and even positions before inventing a single formula.

Common traps include accepting the first plausible next term, ignoring sign changes, and forcing a linear step when the rule is multiplicative.

Under JOA pacing, spend a few seconds classifying the pattern family, then commit or flag — do not re-derive every series from scratch.

Last updated: August 2026

What number series measure on the JOA

Numerical reasoning on the Job Opportunities Assessment (ADF-RECREF132, revised 27 June 2025) includes number series — sequences where you must find the next term, a missing middle term, or the rule that generates the list. These items sit inside the single 51-question / 20-minute assessment alongside verbal and abstract reasoning. They are not a separate Mathematical Ability paper, and they are not a school maths curriculum test.

What the ADF is sampling is practical: can you notice structure in numbers quickly, check a candidate rule against every term, and move on under severe time pressure? A clean series habit is more valuable than memorising advanced techniques you will never need in 23 seconds.

Mental model: classify, verify, answer

Use a three-step loop on every series stem:

Classify the likely family (constant difference, changing difference, multiply/divide, hybrid, alternating).

Verify the rule on all given terms, not just the first pair.

Answer (or flag) once the rule predicts one clear value.

If two rules both seem to fit the given terms but disagree on the blank, you have not verified enough — keep testing or flag and return.

First differences: the default opening move

Write the gaps between consecutive terms (first differences). If those gaps are constant, you have a simple arithmetic series.

Worked example A — constant first difference

Series: 7, 12, 17, 22, ?

  • Step Calculation
  • 12 − 7 +5
  • 17 − 12 +5
  • 22 − 17 +5
  • Next 22 + 5 = 27

That is the easiest family. On the JOA you should solve it in well under the average budget so you bank time for harder stems.

Worked example B — second differences (the step changes by a constant)

Series: 3, 5, 9, 15, 23, ?

First differences: +2, +4, +6, +8

Those differences themselves increase by +2 each time (second difference constant). Next first difference is +10 , so next term is 23 + 10 = 33 .

  • Term First difference Second difference
  • 3 → 5 +2 —
  • 5 → 9 +4 +2
  • 9 → 15 +6 +2
  • 15 → 23 +8 +2
  • 23 → ? +10 +2

Why this matters: many candidates only look for “add the same number.” When the addend itself climbs, they freeze or invent a wrong multiply. Difference tables are faster than algebra.

Negative and decreasing series

The same method works when numbers fall: 40, 33, 26, 19, ? has constant first difference −7 , so the next term is 12 . Sign flips and decreasing runs are deliberate traps for people who always assume growth.

Multiply, divide, and hybrid (multiply-then-add) rules

When first differences jump around with no clean second-difference pattern, test multiplicative structure.

Pure multiply / divide

Series: 2, 6, 18, 54, ?

Each term × 3 : 2×3=6, 6×3=18, 18×3=54, so next is 162 .

Series: 81, 27, 9, 3, ?

Each term ÷ 3 : next is 1 .

Hybrid: multiply then adjust

A very common JOA-style pattern is ×a then +b (or ×a then −b) applied the same way each step.

Worked example C — ×2 + 1

Series: 3, 7, 15, 31, ?

Check: 3×2+1=7, 7×2+1=15, 15×2+1=31, 31×2+1= 63 .

If you only looked at first differences (+4, +8, +16), you might notice doubling differences and still land on +32 → 63 — two routes, same answer. Either route is fine if it is verified on every step .

Worked example D — ×3 − 2

Series: 4, 10, 28, 82, ?

4×3−2=10, 10×3−2=28, 28×3−2=82, 82×3−2= 244 .

How to hunt hybrids quickly

For consecutive terms A → B, ask: “What small integer multiply nearly gets me there?” If 2A is close to B, test ×2 ± something . If 3A is close, test ×3 ± something . Keep the adjuster constant across the whole series before you accept it.

  • Signal in the series Likely family
  • Steady +k or −k Arithmetic (constant first difference)
  • Differences form +2, +4, +6… or similar Second-difference / quadratic-style growth
  • Terms roughly double or triple each time Multiply / hybrid
  • Ratios near 2, 3, ½, ⅓ but not exact Hybrid multiply-add
  • Odd positions one rule, even another Alternating (next subsection)
  • Alternating rules: two series braided into one

Alternating series are high-value traps. The list looks random until you separate odd positions and even positions .

Worked example E — two interleaved arithmetic series

Series: 5, 20, 8, 17, 11, 14, ?

Odd positions (1st, 3rd, 5th, 7th…): 5, 8, 11, ? → +3 each time → next odd-position term 14 .

Even positions: 20, 17, 14 → −3 each time (used only to confirm the braid).

Because the blank is the 7th term (odd position), the answer is 14 . A candidate who forces one rule across all terms might invent nonsense like “add 15, subtract 12…” and pick a distractor.

Worked example F — alternate + and ×

Series: 2, 4, 5, 10, 11, 22, ?

Pattern of operations between terms: ×2, +1, ×2, +1, ×2, +1…

2×2=4, 4+1=5, 5×2=10, 10+1=11, 11×2=22, 22+1= 23 .

Here the alternation is in the operation , not only in separate position tracks. Both ideas matter: split by position or alternate the operator, depending on what fits all given terms.

Position numbering tip

Count carefully. If the stem asks for the next term after six given numbers, that blank is position 7. If it asks for a missing middle term, mark the blank’s index before applying an odd/even split.

Common traps and how to defuse them

  • Trap What it looks like Defence
  • First-pair bias Rule fits terms 1→2 only Verify on every consecutive pair
  • Ignoring signs Treating −4 like +4 Write signed differences
  • Over-algebra Building a complicated formula Prefer simple rules that fit all terms
  • Distractor = intermediate step Option equals a difference, not the term Re-read what the blank is asking for
  • Forcing arithmetic on a geometric series Adding when you should multiply Check ratios when gaps explode
  • Missing the braid One rule fails; give up Split odd/even positions
  • Worked trap example

Series: 2, 3, 5, 8, 12, ?

First differences: +1, +2, +3, +4 → next difference +5 → 17 . A distractor might be 13 (add 1 only at the end) or 20 (double the last term). Partial-pattern distractors are intentional.

JOA process habits for series

Budget: easy constant-difference series should cost ~10–15 seconds; hybrid or alternating items may need ~25–35. Past ~35–40 with no rule → flag and move (see section 4.3).

Scratch lightly: differences and a quick ratio check beat long written algebra.

Response format: many items are multi-choice; some official-style numerical examples use typed numeric answers . Enter the value carefully; do not round unless the stem implies an integer answer you already have.

Goal alignment: the official goal is as many correct attempts as possible. A flagged hard series is better than a three-minute stand-off that costs two verbal items you would have crushed.

  • Mini drill set (self-check)
  • 9, 13, 17, 21, ? → differences +4 → 25
  • 1, 2, 6, 24, ? → ×2, ×3, ×4 → next ×5 → 120
  • 10, 7, 12, 9, 14, ? → odd positions 10,12,14 (+2); even positions 7,9,? (+2); blank is 6th (even) → 11
  • 6, 13, 27, 55, ? → ×2+1 each time → 111

If any of those felt slow, rewrite the difference table until the move is automatic. Series skill on the JOA is less about inventiveness and more about disciplined classification under the clock .