PorismaSource review

Your source-to-article checklist

Source review · Recurrence relations

All source-review topics · Coverage and provenance · Student practice

Every question has a place in the ledger. That does not prove every item is accurate or complete. Use these links to compare the native question, subparts, working, teaching notes, tables and diagrams with your private originals.

“Source comparison recorded” means a visual source-comparison pass is documented. “Mapped” or “generated” describes an ingestion step. Mathematical review, browser interaction tests and settled screenshot review are separate checks.

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13 question records

Optional ticks record your own inspection in this browser only. They do not change the conversion or mathematical-review claims.

13 · Recurrence relations

Edition metadata →

13-01 · Recover terms from a cumulative sumQuestion pp.1 / Solution pp.1

Question and explanations → · Complete native source article →

[2011/ACJC/Promo/13b]. Printed/task labels inventoried: i, ii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

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0 page-item records.

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Tables

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0 page-item records.

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Diagrams

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Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

13 Recurrence relations.pdf · p.1
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.1

13-02 · A telescoping incrementQuestion pp.1 / Solution pp.1

Question and explanations → · Complete native source article →

[2011/DHS/Prelim/II/4b(i),(ii)]. Printed/task labels inventoried: i, ii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

13 Recurrence relations.pdf · p.1
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.1

13-03 · Alternation around a fixed pointQuestion pp.1 / Solution pp.2, 3

Question and explanations → · Complete native source article →

[2023/DHS/Prelim/II/2]. Printed/task labels inventoried: a.i.A, a.i.B, a.ii, b.i, b.ii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

6 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Diagrams

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

13 Recurrence relations.pdf · p.1
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.2
13 Recurrence relation sol.pdf · p.3

13-04 · Shift an affine recurrence into a GPQuestion pp.1 / Solution pp.3, 4

Question and explanations → · Complete native source article →

[2011/TJC/Prelim/II/1)]. Printed/task labels inventoried: i, ii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

13 Recurrence relations.pdf · p.1
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.3
13 Recurrence relation sol.pdf · p.4

13-05 · A rational recurrence with period fourQuestion pp.2 / Solution pp.4, 5

Question and explanations → · Complete native source article →

[2015/TJC/Promo/6]. Printed/task labels inventoried: a, b.i, b.ii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

4 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

The limit substitution must first exclude the pole L=2; dividing by a denominator tending to zero is invalid.

Essential answer conditions and source qualifications

b.ii

$u_1\ne0,1,2$ avoids the intervening poles.

Exact private source page references

13 Recurrence relations.pdf · p.2
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.4
13 Recurrence relation sol.pdf · p.5

13-06 · Sum polynomial and exponential incrementsQuestion pp.2 / Solution pp.5

Question and explanations → · Complete native source article →

[2015/AJC/Promo/11(b)]. Printed/task labels inventoried: none recorded; check the question itself.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

4 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

13 Recurrence relations.pdf · p.2
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.5

13-07 · A logarithmic fixed-point iterationQuestion pp.2 / Solution pp.6

Question and explanations → · Complete native source article →

[2012/MI/Prelim/10]. Printed/task labels inventoried: i, ii, iii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

4 page-item records.

Inspect native target → I compared this with the original

Teacher notes

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

2 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Finding possible limits does not prove convergence; every logarithm requires $x_n$ >1.

Essential answer conditions and source qualifications

Every recurrence step must be defined; limit 1 is impossible.

iii

$x_n>1$ .

Exact private source page references

13 Recurrence relations.pdf · p.2
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.6

13-08 · A square-root iteration crosses its limitQuestion pp.3 / Solution pp.7

Question and explanations → · Complete native source article →

[2012/IJC/Prelim/II/3]. Printed/task labels inventoried: i, ii, iii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

4 page-item records.

Inspect native target → I compared this with the original

Teacher notes

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

The source has a square root over $10-3x_n$ ; plain text extraction loses it. The recurrence is assumed defined and positive at every step.

Essential answer conditions and source qualifications

iii

For the prescribed positive, defined sequence.

Exact private source page references

13 Recurrence relations.pdf · p.3
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.7

13-09 · A rational map approaches twoQuestion pp.3 / Solution pp.7, 8

Question and explanations → · Complete native source article →

[2011/TPJC/Prelim/I/2]. Printed/task labels inventoried: i, ii, iii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

5 page-item records.

Inspect native target → I compared this with the original

Teacher notes

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

1 page-item record. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

13 Recurrence relations.pdf · p.3
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.7
13 Recurrence relation sol.pdf · p.8

13-10 · Halve the error at every stepQuestion pp.3 / Solution pp.8, 9

Question and explanations → · Complete native source article →

[2011/TJC/Promo/14]. Printed/task labels inventoried: a.i, a.ii, a.iii, b.i, b.ii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

6 page-item records.

Inspect native target → I compared this with the original

Teacher notes

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Use $v_{n+1}=v_n/2$ rather than dividing by $v_n$ , which could be zero.

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

13 Recurrence relations.pdf · p.3
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.8
13 Recurrence relation sol.pdf · p.9

13-11 · A conditional limit in a two-cycle mapQuestion pp.3 / Solution pp.9, 10

Question and explanations → · Complete native source article →

[2011/CJC/Prelim/I/2]. Printed/task labels inventoried: i, ii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

1 page-item record. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

The printed question assumes convergence only in (i). Do not infer convergence of every sequence from the one-step inequalities.

Essential answer conditions and source qualifications

The convergence premise forces the constant sequence: F∘F is the identity, so nonconstant admissible sequences form two-cycles.

This one-step comparison does not imply convergence.

Exact private source page references

13 Recurrence relations.pdf · p.3
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.9
13 Recurrence relation sol.pdf · p.10

13-12 · An exponential denominator creates a stable limitQuestion pp.4 / Solution pp.11, 12

Question and explanations → · Complete native source article →

[2011/YJC/Prelim/I/1]. Printed/task labels inventoried: i, ii, iii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

4 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

2 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

The denominator is 2 raised to $x_n$ . Establishing an upper bound is necessary; increasing alone does not imply convergence.

Essential answer conditions and source qualifications

iii

$0<x_1<\alpha$ ; the sequence stays below $\alpha$ .

Exact private source page references

13 Recurrence relations.pdf · p.4
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.11
13 Recurrence relation sol.pdf · p.12

13-13 · An alternating fifth-root iterationQuestion pp.4 / Solution pp.12, 13

Question and explanations → · Complete native source article →

[2014/TJC/Promo/7]. Printed/task labels inventoried: i, ii, iii.a, iii.b, iv.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

6 page-item records.

Inspect native target → I compared this with the original

Teacher notes

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Tables

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

4 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

13 Recurrence relations.pdf · p.4
13 Recurrence relations.pdf · p.5
13 Recurrence relation sol.pdf · p.12
13 Recurrence relation sol.pdf · p.13

An empty omission list is not a completeness certificate. Add any issue you find to the article or page-item record before calling that part finished.