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Quantitative Reasoning 4% exam weight

Series Completion (Numbers and Figures)

Part of the NCEE (National Common Entrance Examination) study roadmap. Quantitative Reasoning topic qr-5 of Quantitative Reasoning.

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Series Completion (Numbers and Figures)

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Series Completion tests how quickly you spot the rule hidden in a chain of numbers, letters, or figures and then write the term that fits. In NCEE Quantitative Reasoning, this sub-section in most keys delivers 3–5 multiple-choice items at roughly 4% of the paper, in many papers inside a 30-minute paper that contains 45–60 questions in total.

The fastest solve path is to compute first differences, then second differences if needed, before guessing the rule. Common families the NCEE recycles: arithmetic progressions (AP), geometric progressions (GP), squares, cubes, triangular numbers, Fibonacci-type recurrence, and alternating two-series.

  • AP check: differences are constant → rule is d.
  • GP check: ratios are constant → rule is r.
  • Alternating check: split odd and even positions into two sub-series and solve each separately.
  • Figure check: track sides, vertices, rotation, or shading step by step.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Core rule families tested

The NCEE draws almost every series item from a small set of rule families. Memorising the look of each family saves time because you stop re-deriving from scratch on every question.

FamilyDetection cluenth-term formula
Arithmetic progression (AP)First differences constanta_n = a + (n−1)d
Geometric progression (GP)Ratios constanta_n = a · r^(n−1)
Quadratic / second-differenceFirst differences increase by a fixed d₂a_n = an² + bn + c
Fibonacci-typeEach term = sum of previous twoa_n = a_{n−1} + a_{n−2}

a = first term, d = common difference, r = common ratio, n = position index. All are unitless.

Step-by-step solving method

A reliable NCEE approach is to apply the same five checks to every item, in the same order, so you never miss a hidden rule.

  1. List the terms and compute first differences Δ₁.
  2. If Δ₁ is not constant, compute second differences Δ₂. Constant Δ₂ ⇒ quadratic rule.
  3. If signs alternate between positive and negative, treat the series as two interleaved APs/GPs and solve each on its own.
  4. If the terms look multiplicative (2, 6, 18, 54 …) compute ratios to test for a GP.
  5. Verify the rule you found against every given term, not just the last two — NCEE distractors punish single-term matches.

Figure-based completion

For figure series, count one attribute at a time: number of sides, vertices, lines of symmetry, rotation angle, or shaded fraction. The rule that fits all given figures wins; a rule that fits only the last figure is almost always wrong.

  • NCEE figure items in most keys change one attribute per step, so do not invent two simultaneous changes.
  • Reflection and 90° rotation are common distractors; always check both before choosing.
  • When shading flips between two colours, split the figure series exactly like an alternating number series.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge cases the NCEE exploits

Most wrong answers on this topic come from mis-classifying the rule, not from arithmetic slips. Watch for these traps:

TrapWhat it looks likeCorrect handling
Mixed AP and GP2, 4, 12, 48, 240 …Each term multiplies by 2, then by 3, alternating → split into two sub-series
Quadratic disguised as AP3, 6, 11, 18, 27 …Second differences are 2, confirming n² + n + 1
Digit-operation rule13, 14, 16, 19, 23 …Differences 1, 2, 3, 4 — another AP hidden inside the differences
Reverse-digit rule11, 22, 44, 88 …GP with r = 2
Near-Fibonacci1, 1, 2, 4, 7 …Not Fibonacci; differences are 0, 1, 2, 3 (another AP)

Worked micro-example

Find the missing term in 3, 6, 11, 18, 27, ?

  1. First differences: 3, 5, 7, 9.
  2. Second differences: 2, 2, 2 → constant, so the series is quadratic.
  3. Next first difference = 9 + 2 = 11.
  4. Missing term = 27 + 11 = 38.

General form: a_n = n² + n + 1, giving 3, 6, 11, 18, 27, 38 — verified across all six terms.

Practice prompts

  1. Identify the rule and the missing term: 2, 6, 12, 20, 30, ? (Hint: triangular-number pattern.)
  2. The figure series shows a square, then the same square rotated 45°, then rotated 45° again. What rotation completes the cycle back to the original orientation?

Exam strategy

Spend no more than 60–90 seconds per series item; longer means the rule is alternating and you must split, not brute-force. On NCEE papers since the syllabus was unified under the National Examination Council (NECO) style, series items cluster in the middle third of the Quantitative Reasoning booklet — finish the easier arithmetic and word problems first, then return.

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