4.1 Number Series Patterns

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:

1. Classify the likely family (constant difference, changing difference, multiply/divide, hybrid, alternating).
2. Verify the rule on all given terms, not just the first pair.
3. 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, ?

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.

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.

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

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)

1. 9, 13, 17, 21, ? → differences +4 → 25
2. 1, 2, 6, 24, ? → ×2, ×3, ×4 → next ×5 → 120
3. 10, 7, 12, 9, 14, ? → odd positions 10,12,14 (+2); even positions 7,9,? (+2); blank is 6th (even) → 11
4. 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.