8.1 Number Sequences

Why number sequences sit in Core Abilities problem solving

The Queensland Fire Department (QFD) Recruit Firefighter Online Cognitive Ability Test, delivered through Pearson VUE / OnVUE, currently names two components: Mechanical Reasoning and Core Abilities. Inside Core Abilities, QFD's April 2024 Firefighter Recruitment Candidate Information Pack names four elements: literacy, numeracy, problem solving, and analytical aptitude. Number sequences belong with problem solving because the skill is not "can you add 7" — that is Chapter 7 numeracy — but "can you find the hidden rule and apply it one more time while a timer runs."

QFD does not publish how many sequence items appear, how they are weighted against literacy or mechanical items, or a percentage cut score. Independent OpenExamPrep practice on the rule types below is useful because timed cognitive batteries of this kind repeatedly use these patterns. That is not a claim that these materials copy unpublished live QFD items.

Pearson OnVUE rules for this sitting ban pens, pencils, note paper, phones, and calculators. You cannot jot a difference table in the margin. The working method in this section is therefore a spoken rule: three to six words you can repeat while you test every given term.

The one diagnostic question

Before you force arithmetic onto a list, name the kind of change:

1. Same amount added or subtracted each step (arithmetic).
2. Same factor multiplied or divided each step (geometric).
3. Two operations taking turns on a single run of numbers (alternating).
4. Two separate sequences sharing odd and even slots (interleaved).
5. First differences that themselves change by a constant (differences of differences).

If the blank is a shape, a rotation, or a matrix cell, you are in Chapter 9 analytical aptitude. Leave those items for that chapter. This chapter stays with numbers.

Arithmetic sequences: constant difference

An arithmetic sequence adds or subtracts the same integer at every step. The common difference does not grow and does not shrink.

Worked Queensland Fire and Rescue (QFR) example. A station's weekly hydrant-inspection counts over five weeks are 12, 15, 18, 21, 24. Each week adds 3 inspections. The next week is 27. The rule is "add 3," not "the numbers sit around 20."

Second example, decreasing. Rural kilometres logged on successive night shifts: 48, 41, 34, 27. The common difference is −7. The next term is 20. If a choice offers 24, someone subtracted 3 from 27 (chasing the last two digits) instead of subtracting the common difference.

OnVUE check. Subtract consecutive pairs in your head: 15 − 12, 18 − 15, 21 − 18. If those three results are the same, you have an arithmetic rule. If they are not the same, stop. Do not force "add 3" onto a sequence that only started that way.

Geometric sequences: constant ratio

A geometric sequence multiplies or divides by the same factor each step. The common ratio is what you hold, not a difference.

Worked example. Spare breathing apparatus (BA) sets staged at a district cache over four restock days: 3, 6, 12, 24. Each term is twice the previous term. The next term is 48. The first differences look like +3, +6, +12 — that is a clue that this is not arithmetic. Differences that themselves double are often a geometric sequence in disguise.

Second example, dividing. Foam 20-litre drums remaining in a locker: 81, 27, 9, 3. Each term is divided by 3. The next term is 1. A choice of 0 is a remainder-style guess, not the geometric rule.

OnVUE check. Ask whether term 2 ÷ term 1 equals term 3 ÷ term 2. For 3, 6, 12 that is 2 and 2. For 2, 4, 8, 14 the ratios are 2, then 2, then 14 ÷ 8 = 1.75. Abandon doubling. 14 is a trap built for people who added 6 after two successful doubles.

Alternating sequences: two operations, one line

An alternating sequence applies two operations in turn to the same run of numbers. There is still one sequence, not two hidden lists.

Worked example. Recorded turnout times in minutes: 10, 14, 13, 17, 16, 20. Walk the operations: 10 + 4 = 14, 14 − 1 = 13, 13 + 4 = 17, 17 − 1 = 16, 16 + 4 = 20. The pair is +4 then −1. The next term is 20 − 1 = 19.

Second example. Lengths of 30 m hose moved from the rack over six jobs: 5, 10, 8, 16, 14, 28. Operations: ×2, −2, ×2, −2, ×2. The next operation is −2, so 28 − 2 = 26. A choice of 56 assumes you keep doubling and ignore the subtraction that has already appeared twice.

The trap is to average the whole list, or to treat the last three terms as a small arithmetic piece. Test the pair of operations against every step, including the last given step. If ×3 then −3 fits 7, 21, 18, 54, 51, 153, the next step is −3 and the next term is 150 — not another ×3.

Two interleaved sequences: odd slots versus even slots

Sometimes the list is two sequences zipped together. Odd-numbered positions follow one rule; even-numbered positions follow another.

Worked example. 4, 40, 7, 35, 10, 30, ?
Odd positions: 4, 7, 10 — arithmetic +3. The 7th term (odd) is 13.
Even positions: 40, 35, 30 — arithmetic −5. If the blank had been the 8th term, it would be 25.

Second example. 2, 100, 4, 90, 8, 80, 16, ?
Odds: 2, 4, 8, 16 — geometric ×2.
Evens: 100, 90, 80 — arithmetic −10. The blank is the 8th term (even), so 70. A choice of 32 is the next odd term, which is the wrong slot.

OnVUE move. When three arithmetic or geometric checks fail, read every second number. If those look clean, you have an interleaved item. Count positions on your fingers if you must — you still cannot use paper. The blank inherits only the rule of its own slot.

Differences of differences

When first differences are not constant, compute the differences of those differences. A constant second difference means the next first-difference continues that first-difference arithmetic sequence.

Worked example. Incident reports closed per day: 2, 5, 10, 17, 26.
First differences: +3, +5, +7, +9.
Second differences: +2, +2, +2.
The next first difference is +11. The next term is 26 + 11 = 37.

Second QFR example. Pump-test figures recorded on successive days: 3, 6, 11, 18, 27. First differences +3, +5, +7, +9; second differences +2. Next first difference +11; next term 27 + 11 = 38.

You do not need the closed algebraic form (often a quadratic such as n² + 1). Extending the difference table in your head is enough, and it is what OnVUE actually allows. Whisper the first differences, confirm the second differences are constant, then add the next first difference to the last given term — forgetting that last addition is a common miss even after a correct difference table.

Holding the rule with no paper

OnVUE will not let you draw a difference table. Use this order:

1. Say a candidate rule in a short phrase ("add 4", "times 3 then minus 3", "odds add 3").
2. Test that phrase on every given step, not only the first pair.
3. If it fails at step three, drop it. Do not patch it with a new exception.
4. If simple checks fail, split odd and even positions.
5. If first differences change smoothly by the same extra amount, try second differences.
6. If twenty seconds produce no rule, skip and review later if the platform allows. The Candidate Information Pack states that candidates are not expected to answer every question correctly.

Traps that cost an otherwise easy item

• Matching only the first two terms, then ignoring a later contradiction.
• Calling 4, 8, 16, 20 geometric because it "started by doubling."
• Computing the next difference and then forgetting to add it to the last given term.
• Filling the blank with the next term of the wrong interleaved slot.
• Practising shape-sequence drills in this chapter; those belong with analytical aptitude.

The flowchart below is the decision order to run in your head on a live item.

Run that order once. Do not restart at arithmetic after you have already seen the ratio fail, and do not jump to second differences before you have checked interleaved slots — two zipped arithmetic lists can mimic a messy difference table if you never split the positions.