Series Completion (Numbers)

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Series Completion (Numbers) asks you to spot the rule governing a given numeric sequence and pick the next (or missing) term. The fastest method is to compute first differences between consecutive terms: if they are constant, the series is an Arithmetic Progression (AP) with a_n = a + (n-1)d. If the first differences themselves form an AP, the underlying pattern is quadratic (n²-type), and a constant second difference signals a cubic-style rule.

For multiplicative patterns, test the ratio between consecutive terms; a constant ratio indicates a Geometric Progression (GP) with a_n = a · r^(n-1). Also recognise squares (n²), cubes (n³), triangular numbers n(n+1)/2, Fibonacci (F_n = F_{n-1} + F_{n-2}), prime sequences, and alternating +/− sub-series.

Compute differences before guessing squares and cubes.
Split alternating series into odd-indexed and even-indexed sub-sequences.
Always verify your candidate rule against every given term, not just the last two.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Pattern-Detection Workflow

Begin by listing the terms, then compute successive first differences. A flat row of differences confirms an AP; a row that itself grows linearly points to a quadratic rule. Only after these quick checks should you attempt ratio analysis, because a near-GP in many papers hides a polynomial pattern. NAT-I (NTS) in standard papers delivers 4–6 term sequences, giving you enough data points to reject a wrong rule early.

Alternating and Interleaved Series

When signs flip or magnitudes jump irregularly, split the series into two sub-series at odd and even indices. Each sub-series is in most keys a simple AP or GP. Failing to separate them is the most common reason students extend the wrong branch and pick a distractor answer.

Rule Families at a Glance

Pattern family	Tell-tale signature	Example
AP	Constant first difference	3, 7, 11, 15, 19 (d = 4)
GP	Constant ratio	2, 6, 18, 54, 162 (r = 3)
n² series	First differences form AP	1, 4, 9, 16, 25
n³ series	Constant second difference	1, 8, 27, 64, 125
Triangular	n(n+1)/2	1, 3, 6, 10, 15
Alternating AP	Odd/even index splits	5, 12, 8, 15, 11, 18

Common trap: 2, 6, 18, 53 looks like a GP with r = 3, but the 53 breaks it — the actual rule is quadratic. Always fit the rule to all terms before committing.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Advanced Patterns

Beyond standard AP/GP, NAT-I (NTS) occasionally features digit-sum patterns (e.g., 18, 27, 36 whose digits sum to 9, 9, 9), digit-reversal rules, and mixed polynomials such as n² + n (yielding 2, 6, 12, 20, 30). Composite patterns like n² + 1 produce 2, 5, 10, 17, 26 — easily misread as AP at first glance. Cubes in disguise (64, 125, 216 = 4³, 5³, 6³) trap students who instinctively reach for square-based rules.

Worked Micro-Example

Consider the sequence 2, 5, 10, 17, 26, ?. First differences: 3, 5, 7, 9 — an AP with d = 2. Second differences are constant at 2, confirming a quadratic rule a_n = n² + 1. The next first difference must be 11, so the missing term is 37. Verification: positions 1–5 give 2, 5, 10, 17, 26 — all match.

Common Mistakes and Practice Prompts

Mistake	Fix
Continuing the wrong sub-series in alternating sequences	Split odd/even indices first
Treating near-GP as true GP	Verify ratio on every pair
Using only the last two terms	Check rule against all given terms
Confusing cubes with squares	List n² and n³ tables during revision

Practice 1: Find the next term of 3, 6, 11, 18, 27, ? Practice 2: Identify the rule in 1, 2, 4, 7, 11, 16, ? and give the next term.

Exam strategy: budget 45–60 seconds per question, and if no pattern emerges in 30 seconds, skip and return — the 4% weight doesn’t justify a time sink.

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Series Completion (Numbers)