HSC Mathematical Induction: Inequalities, Divisibility, and Series Proofs

 

Intro

The HSC Induction booklet is a free Mathematics Extension 2 resource with 54 worked problems covering weak and strong induction, structural induction on algebraic forms, inequalities, and combinations with algebra, recurrences, and integrals. It is written for Extension 2 students aiming to deepen proof-writing through induction and designed for structured HSC revision on Vu's Maths Hub.

This deep-dive introduces HSC Induction — browser-readable, aligned with the NESA syllabus.

Summary

The HSC Induction booklet offers a fundamentals review, 54 tiered problems with solutions, appendices, and a conclusion that distils exam habits. Open HSC Induction — no account required. Use this post to plan how to work through a 41-page booklet efficiently.

What is this booklet?

This booklet is written for Extension 2 students aiming to deepen proof-writing through induction.

Focus: weak and strong induction, structural induction on algebraic forms, inequalities, and combinations with algebra, recurrences, and integrals.

Topics covered:

• Mathematical induction
• Weak and strong induction
• Structural induction
• Inductive proofs for inequalities
• Combinations with algebra and recurrences
• Integral-flavoured induction (enrichment)

How to use it:

• Read overview and induction primer first
• Part 1 without hints
• Part 2 hints only after genuine attempt
• Revisit and re-derive proofs without notes

Approximately 41 pages, CC BY 4.0, readable at HSC Induction.

Key fundamental reviews

Core ideas are embedded in the introduction and early problems. Before Part 1, ensure you can handle the following without notes:

• Induction primer: base case, inductive step, conclusion
• Weak vs strong induction — when each applies
• Summation and divisibility templates
• Inequality proofs with monotonicity
• Structural induction on expressions
• Connecting induction to recursive definitions

Why fundamentals matter

Induction primer — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Weak vs strong induction — when each applies — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Summation and divisibility templates — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Inequality proofs with monotonicity — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Structural induction on expressions — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Students who skip this section often repeat the same algebra errors in Part 2. Treat HSC Induction fundamentals as a closed-book quiz first.

Problems and how to use them

The HSC Induction booklet packs 54 practice problems into roughly 41 pages — well beyond a single textbook chapter. Each item includes worked solutions; many include Takeaways that highlight the method to reuse in exams.

Overall structure

Part 1 — detailed solutions: basic, medium, and advanced induction proofs with detailed solutions.

Part 2 — hint-based fluency: matching tiers with hints only after attempt.

Use Part 1 to learn how complete NSW HSC working is written. Use Part 2 in the fortnight before trials — hints are upside-down so you attempt first.

Other (10 problems)

05 (1 problems)

This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Recursive sequence via f(x)=2^x+2-x" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Recursive sequence via f(x)=2^x+2-x

06 (1 problems)

This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Weighted geometric sum" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Weighted geometric sum

07 (1 problems)

This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Product of odd factorials vs powers of factorials" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Product of odd factorials vs powers of factorials

08 (1 problems)

This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Bounding sums of cubes" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Bounding sums of cubes

09 (1 problems)

This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Product of odd factorials inequality" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Product of odd factorials inequality

10 (1 problems)

This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Bounding cubic sums" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Bounding cubic sums

11 (1 problems)

This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Reduction of the Tangent Integral" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Reduction of the Tangent Integral

12 (1 problems)

This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Logarithmic Reduction and Closed Form" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Logarithmic Reduction and Closed Form

13 (1 problems)

This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Sum of consecutive odd numbers" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Sum of consecutive odd numbers

14 (1 problems)

This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Logarithmic derivative of a polynomial" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Logarithmic derivative of a polynomial

Part 1 (15 problems)

Advanced (5 problems)

This tier contains 5 problems aimed at extension and synthesis. Representative work includes "De Moivre's Theorem" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• De Moivre's Theorem
• Tiling a Defective 2^n \times 2^n Board
• Evaluating I_n from a Recurrence
• Integer Coefficients in \int_0^1 x^n e^x , dx
• Closed Form for J_n

Basic (5 problems)

This tier contains 5 problems aimed at foundational fluency. Representative work includes "Telescoping Harmonic Sum" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Telescoping Harmonic Sum
• Divisibility of n^3+2n
• Nine divides 7^n + 2^n
• Triangular Numbers
• Polygon Interior Angle Sum

Medium (5 problems)

This tier contains 5 problems aimed at exam-standard reasoning. Representative work includes "Bounding a Basel-type Sum" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Bounding a Basel-type Sum
• Comparing n! and 2^n
• Solving a Linear Recurrence
• Derivative of x^n
• Factorising x³n-1

Part 2 (29 problems)

Advanced (14 problems)

This tier contains 14 problems aimed at extension and synthesis. Representative work includes "Binomial Theorem" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Binomial Theorem
• Fermat's Little Theorem
• Symmetric Sum Inequality
• Tower of Hanoi
• Vandermonde's Identity
• Wilson's Theorem
• Recursive Sequence with Surds
• Powers of 2+3
• Sum of Cosines
• Nested Radicals
• Cosecant Sum Formula
• Arctangent Sum
• Fermat Number Products and Series
• Advanced: Fibonacci Telescoping Series

Basic (7 problems)

This tier contains 7 problems aimed at foundational fluency. Representative work includes "Sum of First n Odd Numbers" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Sum of First n Odd Numbers
• Divisibility of 4^n - 1 by 3
• Sum of First n Squares
• Geometric Series
• Bernoulli's Inequality
• De Moivre's Formula (Conjugate Form)
• Power Inequality 2^n \ge n^2 - 2

Medium (8 problems)

This tier contains 8 problems aimed at exam-standard reasoning. Representative work includes "Sum of Cubes" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

• Sum of Cubes
• Tower Inequality
• Inequality with Factorials
• Postage Stamp Problem
• Inequality 3^n > n^3
• Factorial Inequality (2n)! \ge 2^n (n!)^2
• Exponential Inequality 4^n - 1 - 7n > 0
• Weighted Geometric Sum

Common patterns across the booklet

• Integration: 2 problems — e.g. "Reduction of the Tangent Integral"
• Mechanics: 1 problem — e.g. "Telescoping Harmonic Sum"
• Inequalities: 12 problems — e.g. "Bounding sums of cubes"
• Probability & counting: 1 problem — e.g. "Binomial Theorem"
• Polynomials: 3 problems — e.g. "Product of odd factorials vs powers of factorials"
• Trigonometry: 5 problems — e.g. "Polygon Interior Angle Sum"

Standout and less-seen problem types

These go beyond routine drills — expect unfamiliar wording or multi-topic synthesis:

• De Moivre's Theorem: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
• Tiling a Defective 2^n \times 2^n Board: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
• Evaluating I_n from a Recurrence: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
• Integer Coefficients in \int_0^1 x^n e^x , dx: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
• Closed Form for J_n: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
• Binomial Theorem: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.

Working through a large booklet

1. Read overview and induction primer first
2. Part 1 without hints
3. Part 2 hints only after genuine attempt
4. Revisit and re-derive proofs without notes

Timed practice: allow about one minute per mark; write legible structure even when practising alone. Error log: tag mistakes as concept, algebra, or misreading. Rotation: do not camp on advanced tier if basics still slip.

Open HSC Induction and work steadily — 54 problems is a marathon, not a sprint.

Key appendices

This booklet has no separate appendix files — formulas and takeaways are embedded in solutions and the conclusion.

Use the conclusion section as a pre-exam checklist and keep your class formula sheet nearby while practising.

Key conclusion

Induction is a core Extension 2 reasoning tool. Mastery comes from repeated reflective practice — sharpen hypothesising, algebraic manipulation in the inductive step, and complete proof communication.

The booklet's closing section reinforces these habits:

• Read overview and induction primer first
• Part 1 without hints
• Part 2 hints only after genuine attempt
• Revisit and re-derive proofs without notes

Revisit HSC Induction in the fortnight before trials and redo problems you missed on first pass.

How to study with this booklet

Pair with Proofs booklet; one induction proof daily for three weeks

General principles:

• Closed-book first: attempt without notes, then check fundamentals.
• Error log: record concept vs algebra vs reading errors.
• Spaced repetition: redo missed questions after 3 and 7 days.
• Past papers last: fix weak topics here, then sit full papers timed.

Mini-FAQ

Who is the HSC Induction booklet for?

Extension 2 students aiming to deepen proof-writing through induction studying Mathematics Extension 2 under the NSW HSC.

Should I read solutions before attempting problems?

Attempt Part 1 first. Use Part 2 hints only after a genuine try or partial working.

Where can I read the booklet online?

Open HSC Induction on Vu's Maths Hub — free, no account required.

How many problems are in the booklet?

Roughly 54 practice problems across 41 pages, each with worked solutions.

Is this aligned with NESA?

Topics match Mathematics Extension 2 outcomes for weak and strong induction, structural induction on algebraic forms, inequalities, and combinations with algebra, recurrences, and integrals. Confirm scope with your teacher and current NESA documentation.

Common mistakes to avoid

• Missing the base case or proving the wrong starting $n$
• Assuming $P(k)$ without using it in the $n=k+1$ step
• Weak algebra when expanding $(k+1)$ terms
• Using induction when a direct closed form is available
• Rushing to advanced tiers before basic fluency — build foundations first.

Practice on Vu's Maths Hub

Open the free HSC Induction on Vu's Maths Hub — 54 problems with full worked solutions.

Related resources:

• How to use Vu's Maths Hub — Study pathway for proof-heavy topics
• HSC Proofs — Proof structure alongside induction

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All booklets are free for personal and school use under the CC BY 4.0 licence.

Related resources:

• HSC Inequalities — Inductive inequality proofs
• HSC Sequences — Series induction

Mathematical induction — exam context

In NSW Mathematics Extension 2 examinations, mathematical induction routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Induction booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Induction and locate items that stress mathematical induction; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Weak and strong induction — exam context

In NSW Mathematics Extension 2 examinations, weak and strong induction routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Induction booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Induction and locate items that stress weak and strong induction; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Structural induction — exam context

In NSW Mathematics Extension 2 examinations, structural induction routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Induction booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Induction and locate items that stress structural induction; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Inductive proofs for inequalities — exam context

In NSW Mathematics Extension 2 examinations, inductive proofs for inequalities routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Induction booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Induction and locate items that stress inductive proofs for inequalities; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Combinations with algebra and recurrences — exam context

In NSW Mathematics Extension 2 examinations, combinations with algebra and recurrences routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Induction booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Induction and locate items that stress combinations with algebra and recurrences; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Integral-flavoured induction (enrichment) — exam context

In NSW Mathematics Extension 2 examinations, integral-flavoured induction (enrichment) routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Induction booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Induction and locate items that stress integral-flavoured induction (enrichment); attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Syllabus alignment

This booklet supports Mathematics Extension 2 under the NESA syllabus. It supplements school instruction with 41 pages of extra exam-style practice — not a replacement for class teaching.

Additional exam advice

When sitting Mathematics Extension 2 exams, allocate time proportional to marks. Practise concise justification in HSC Induction — NSW markers reward clear communication. Reread the booklet conclusion the night before for a habit checklist.

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