Deep Dive: Geometric Series

Provenance: written 2026-09-29. Read in full before writing: research/veridian/WMA12-verified-facts.md (all sections, with a targeted re-grep of every "geometric" hit — §2 spec text, §3 formula-booklet page, §4 mark-scheme conventions, §6.4, §7, and both independent geometric-series.ts audit entries in §9, PRs #449 and #444); content/veridian/curriculum/wma12/geometric-series.ts (the full lesson file, all blocks); content/veridian/technique-hub/wma12/codex-formula-and-method-bank.md, forge-exam-technique-guide.md and signal-frequency-and-boundaries.md (all three, checked for overlap); content/veridian/question-bank/wma12.ts (checked — contains no geometric-series items yet, so nothing to cross-reference there). No Pearson textbook was consulted for this document, consistent with the facts bank's own sourcing discipline.

#CODEX Deep-Dive — Geometric Series (WMA12, spec 4.4)

#What this document is, and isn't

The lesson file (geometric-series.ts) already teaches this topic in full, with a real bookwork-proof derivation, a worked chain, a chain-drill, a method comparison, a marked solution with a scored wrong path, seven trap-taxonomy entries and six MCQs — all of it grounded and, per two independent audit passes, already correct against the real mark schemes. This document does not re-teach any of that, and does not reproduce its worked numbers. What it adds: the complete theory laid out as a single dense reference rather than a teaching narrative; two real sub-cases the lesson's own asides name but never work through (an even-power two-term gap, and the mixed-series real-world context by name); a side-by-side comparison of the arithmetic and geometric bookwork proofs' real mark-scheme shapes, which no existing document in this course states explicitly; and one table synthesising every real exam angle this specific topic has actually been tested from, with its citation. Where the real material runs out, a section says so rather than filling the gap.

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#1. The complete real model

#1.1 Definition and the diagnostic that tells it apart from arithmetic

A geometric sequence is generated by $u_{n+1} = r\,u_n$ — multiply by a fixed common ratio $r$ each step, contrasted with arithmetic's $u_{n+1} = u_n + d$. Given a list of terms, compute consecutive differences and consecutive ratios: whichever comes out constant names the sequence type. This is mechanical, not conceptual, which is exactly why the facts bank names "reaching for the wrong series type's formula mid-question" as a recurring, named error category confirmed across at least two series (June 2019; October 2020, per §6.4).

#1.2 The $n$th term

Unrolling the recurrence from $u_1 = a$ gives $u_n = ar^{n-1}$ — printed in the formula booklet (verified against the rendered P2 booklet page, §3), so this is look-up, not memorise.

#1.3 The finite sum, and the real bookwork proof

$S_n = \dfrac{a(1-r^n)}{1-r}$, $r \neq 1$ — also printed in the booklet. The proof is not, and the spec's own guidance for item 4.4 states plainly that "the proof of the sum formula for a finite series should be known." Multiply $S_n$ by $r$; this shifts every term to the next one, so $rS_n$ shares all but two terms with $S_n$; subtracting collects the two survivors ($a$ and $ar^n$) and cancels the rest; factorise and divide by $(1-r)$.

The real mark-scheme shape, verified, and how it differs from arithmetic's. The facts bank's own independent audit (PR #444) re-extracted the primary mark scheme for this exact proof from October 2020, Q8(a): M1 ("Writes down at least 3 correct terms of a geometric series and multiplies their sequence by $r$") / A1 ("Obtains $S_n - rS_n = a - ar^n$ … both sides must be seen unfactorised") / A1* ("Factorises both sides and divides by $1-r$ to obtain the printed answer") — 3 marks. A separate audit pass (PR #441) independently re-extracted the parallel arithmetic proof's real scheme from January 2020, Q8(i): B1 (an expression for $S$ in $\geq 3$ terms, first and last correct) / M1 (reverse and add, $\geq 2$ matching pairs) / A1* (reach the given result via the intermediate $2S = n\{2a+(n-1)d\}$ line) — also 3 marks. Laid side by side, both proofs cost exactly 3 marks, but not the same way:

The practical consequence, confirmed by the same audit: a candidate who assumes "write $S_n$" is always worth a free mark, because it is on the arithmetic proof, is wrong for the geometric one — nothing is earned for that line alone. Two further, distinct real proof errors are independently confirmed on this exact October 2020 question, not derived: subtracting the wrong way round, producing $S_n - rS_n = ar^n - a$ or $S_n - rS_n = a + ar^n$; and importing the arithmetic proof's reverse-and-add move wholesale, which the examiner records directly — "they wrote the terms with increasing powers of $r$ and then wrote them as decreasing powers of $r$ and then tried to fudge the answer." Both are already built into the lesson's trap taxonomy.

#1.4 Sum to infinity and the convergence condition

$S_\infty = \dfrac{a}{1-r}$ for $|r| < 1$ — the third booklet formula, and not a new argument: it is $S_n$ watched as $n \to \infty$. The only $n$-dependent piece of $S_n$ is $r^n$, so convergence is entirely a question of what $r^n$ does. $|r| < 1 \Rightarrow r^n \to 0 \Rightarrow S_n \to \frac{a}{1-r}$. The condition is on size, not sign — a negative $r$ with $|r|<1$ converges exactly as a positive one does.

#1.5 Degenerate and boundary cases

Two values of $r$ break the formula in different ways, and both are required by the spec's own strict inequality rather than a house convention:

• $r = 1$: the derivation divided by $1-r$, which is illegal here. The series is $a+a+a+\cdots$, summed directly (not via the formula) as $S_n = na$, and it never converges for $a \neq 0$.
• $r = -1$ (the boundary of $|r|<1$ itself): $r^n$ alternates $-1,1,-1,1,\ldots$ forever. For $a=7$: partial sums run $7,0,7,0,\ldots$ — visiting two values endlessly, settling on neither. This is why the specification's inequality is strict ($|r|<1$, not $|r|\leq 1$): the boundary genuinely fails, and in two structurally different ways at $r=1$ and $r=-1$.

The facts bank records no examiner-verified error specifically on this convergence condition for WMA12; the lesson's own two related trap-taxonomy entries (reading $|r|<1$ as "$r<1$"; wrongly admitting $r=-1$) are explicitly marked in that file as derived from the mechanism, not cited to a real question, and this document does not upgrade that status.

#1.6 The two algebraic forms of $S_n$

$\dfrac{a(1-r^n)}{1-r} \equiv \dfrac{a(r^n-1)}{r-1}$ — multiply top and bottom by $-1$. The booklet prints the first. When $r>1$, the first form puts a negative over a negative; the second keeps both positive. Mark schemes' own oe ("or equivalent") convention treats the two as interchangeable; the only real cost of the "wrong" one is an extra sign to track under time pressure, not a lost mark.

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#2. Real sub-cases the archive requires, each with a short micro-example

Every number below is VERIDIAN-original and independently checked; none is taken from a real paper, and none duplicates a number used in the lesson file.

#2.1 Given two non-consecutive terms — divide to eliminate $a$

Two given terms of a geometric sequence are two equations in $a$ and $r$; dividing eliminates $a$ because it appears as a multiplicative factor in both (the direct structural mirror of subtracting to eliminate $a$ in the arithmetic case, where it appears additively). This is a real, confirmed question shape: the facts bank's own audit names October 2020, Q8(b)/(c) as using the 2nd and 5th terms of a geometric sequence — a gap of 3 — bundling "form equations, divide, cube-root" into a single method mark.

Micro-example, odd gap (mirrors the real question's structure). $u_2 = 6$, $u_5 = -48$. Then $r^3 = \frac{ar^4}{ar} = \frac{-48}{6} = -8$, so $r = -2$ (a real cube root is unique). Then $a = \frac{6}{r} = -3$. Check: $ar^4 = -3 \times 16 = -48$. ✓

Micro-example, even gap — a case the lesson's own worked chain flags but never shows. The lesson's chain explicitly notes, of its own $r^3$-gap example, that "an even power would have given two [solutions], and both would have needed considering" — a real caveat, left unworked. Filling it: $u_2 = 8$, $u_4 = 18$. Then $r^2 = \frac{18}{8} = \frac{9}{4}$, so $r = \pm\frac{3}{2}$ — genuinely two values, both real. For $r=\frac32$: $a = \frac{8}{3/2} = \frac{16}{3}$. For $r=-\frac32$: $a = \frac{8}{-3/2} = -\frac{16}{3}$. Both pairs satisfy both given terms; without a further constraint in the question (e.g. "all terms are positive," which would eliminate the negative root), both are mathematically valid and a candidate would need to state both or select on the stated condition.

#2.2 Sigma notation — the same series, differently dressed

Already the lesson's own fixed gap (per the §9 audit), grounded in the real January 2024, Q5(i) mark scheme, which explicitly warns that one substituted term is "not sufficient evidence" of a geometric sequence, and separately credits writing out the first three terms as an alternative route to the same mark.

Micro-example. $\sum_{k=1}^{\infty} 8\left(\frac{1}{3}\right)^k$. Substituting $k=1,2$: first term $\frac{8}{3}$, second term $\frac{8}{9}$; ratio $\frac{8/9}{8/3} = \frac13$, confirmed against a third term ($\frac{8}{27}$, ratio $\frac13$ again). $S_\infty = \dfrac{8/3}{1-1/3} = \dfrac{8/3}{2/3} = 4$.

#2.3 Summing the terms after an exclusion

Confirmed as a real, examiner-verified trap site, not merely a plausible variant: the January 2024 examiner report records candidates "often used $a=6$ instead of $a=1.5$" when asked for the sum of a series after its first term had been removed — taking the sequence's original first term rather than the first term of the sum actually being asked for.

Micro-example. $a = 10$, $r = 0.2$. Whole sum: $S_\infty = \frac{10}{0.8} = 12.5$. First three terms: $10, 2, 0.4$, summing to $12.4$. Terms after the third: $12.5 - 12.4 = 0.1$. Direct check: the 4th term is $10(0.2)^3 = 0.08$, and $\frac{0.08}{1-0.2} = 0.1$. ✓ The two routes agreeing is the self-check this exact trap makes necessary.

#2.4 Mixed arithmetic-then-geometric transition — including its real context

January 2022, Q(c)/(d): an arithmetic series transitions into a geometric one starting from its final term. This paper's own §6.4 commentary calls it "the paper's biggest discriminator that series," but the facts bank's own later audit (§9, PR #444) checks that framing against the primary source directly and finds it looser than the evidence supports: the real examiner report calls it only "the first of three discriminating questions at the end of the paper" (mean score ≈3.4/9) — genuinely discriminating, but not confirmed as the single biggest. This document uses the corrected, narrower claim. The facts bank's own §9 audit additionally confirms this real question sits inside a physical, real-world context — a metal post driven into the ground, its resistance modelled by the two series in sequence (context named but, per the lesson file's own design choice, not reproduced). The two confirmed traps are distinct: double-counting the shared "overlap" term in both sums, and wrongly restarting the geometric part from the whole sequence's original first term rather than from where the arithmetic part actually left off.

Micro-example. Arithmetic: $a=50$, $d=-4$, 6 terms ($50,46,42,38,34,30$); $S_6 = \frac{6}{2}(2(50)+5(-4)) = 240$. From the 6th term onwards, geometric with $r=0.2$: the geometric first term is $30 \times 0.2 = 6$ (not $30$ itself), so its tail sums to $\frac{6}{0.8} = 7.5$. Correct total: $240 + 7.5 = 247.5$. The double-count error instead treats $30$ as the geometric first term: $\frac{30}{0.8} = 37.5$, giving a wrong total of $277.5$ — exactly $30$ too high, one full copy of the shared term, matching the shape of the real error named above.

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#3. Synthesis — every real exam angle this topic has actually been tested from

That is the complete set of geometric-series angles this course's own research has traced to a named, real question. Two structural points fall out of the table rather than any single row: every confirmed trap sits at a transition — between series types, between notations, or between "the sequence" and "the sum actually asked for" — never inside the mechanical application of a single given formula to a single given $a$ and $r$; and every one of the five question-bearing series above (Oct 2020, Jan 2022, Jan 2024, plus the two general-confusion citations, Jun 2019 and Oct 2020) falls within the 17 series the facts bank confirms an examiner report exists for (§5), so none of this synthesis rests on mark-scheme text alone without commentary to corroborate it.

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#4. Frequency, stated honestly

This paper's own SIGNAL document (signal-frequency-and-boundaries.md) counts geometric series as confirmed via an individually-named question in 2 of the 20 series with a question paper and mark scheme reviewed (Jan 2022; Jan 2024) — explicitly stated there as a floor, not a rarity signal. Reading the facts bank's own later audit log (§9, PR #444) directly adds a further, independently-confirmed anchor not folded into that floor at the time it was written: October 2020, Q8, a dedicated finite-sum-proof-plus-two-terms question. Taken directly from the facts bank's own citations, this topic is therefore confirmed via a dedicated question in at least 3 of the 20 series checked (Oct 2020, Jan 2022, Jan 2024), with two further series contributing a citation of a different shape — Jan 2020 (examiner advice naming the geometric proof alongside the arithmetic one, on an arithmetic question) and June 2019 (a general formula-confusion finding, no specific sub-part named). No percentage or likelihood is stated anywhere in this section, in line with the standing rule against inventing one; "3 of 20" (or "5 of 20" counting the two looser citations) is exactly what it says — a floor on what has been individually verified, nothing more.

#5. What this document does not, and cannot, claim

• No real-world-context geometric question beyond the Jan 2022 one is confirmed for WMA12 in the reviewed archive — no compound-interest, depreciation, or population-growth framing was found in any of the citations checked for this document. This is an absence in what has been verified, not a claim that none exists in the full 20-series archive; a targeted re-check would be needed before saying more.
• No "find the least $n$ such that $S_n$ exceeds a value" (logarithm-based inequality) question is confirmed for this paper. This is a familiar geometric-series exam shape on other specifications, but nothing in the facts bank's WMA12 citations places it on this paper, so it is left out of §2 and §3 rather than assumed by analogy.
• Grade-boundary or likelihood figures are deliberately absent from this document. Per the standing rule, this document states only what has actually been seen, phrased as "N of M series checked" where that is available (§4) — never a percentage, and never a boundary mark, which in any case is a whole-paper figure with no topic-level breakdown in any source checked.

#Sources

research/veridian/WMA12-verified-facts.md (§2 spec text; §3 formula booklet; §4 mark-scheme conventions; §5 archive scope, 20/17 series; §6.4; §7; §9's two independent geometric-series.ts audit passes, PRs #449 and #444) — the primary source for every cited fact above. content/veridian/curriculum/wma12/geometric-series.ts, read in full, for what already exists and to avoid duplicating its worked numbers. content/veridian/technique-hub/wma12/codex-formula-and-method-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md, read in full, for what this document does not need to repeat.