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HSC Mathematical Induction booklet — patterns, inequalities, divisibility, and proof practice for Extension 2. Read free on Vu’s Maths Hub.

📚 Study Guide & Outline

1. HSC-Induction - an Introduction

Mathematical Induction is a powerful technique for proving that a statement is true for all positive integers. Across the HSC Mathematics Extension 1 and Extension 2 courses, it serves as the ultimate bridge between algebra and formal proof. Students historically struggle with Induction because it is often their first exposure to rigorous mathematical logic—it requires setting up a foundational base case and conceptually understanding the "domino effect" of the inductive step. While basic divisibility proofs can be memorized, the HSC frequently tests harder inequality proofs or calculus-based induction (like proving a derivative formula for $x^n$ ) which demand true algebraic agility.

2. Learning Outcomes & Syllabus Mapping

Proof by Mathematical Induction.
Prove sums of series and divisibility properties using induction.
Prove inequalities using induction (a major Extension 2 focus).
The Nature of Proof (Induction with first principles or De Moivre's Theorem).

3. Prerequisites

Strong foundational algebra (expanding, factoring, fractions).
Familiarity with sigma ( $\Sigma$ ) notation and series.
For Extension 2: Knowledge of Complex Numbers (De Moivre's Theorem) and Calculus.

4. Common HSC Mistakes

The most frequent mistake is failing to explicitly state the inductive assumption (Step 2) or failing to show exactly where that assumption is substituted into the inductive step (Step 3). Examiners heavily penalize "fudging"—skipping the algebraic steps that logically connect the assumption to the required proof. Forgetting the concluding statement is also a common way to leave a proof frustratingly incomplete.

5. Sample Worked Problem

Question: Prove by mathematical induction that for all integers

n \ge 1

, the sum of the first

n

odd numbers is

n^2

Solution:

Let $P(n)$ be the proposition:

$1 + 3 + 5 + \dots + (2n - 1) = n^2$

Step 1 (Base Case): For $n = 1$ : $\text{LHS} = 1$ $\text{RHS} = 1^2 = 1$ Since $\text{LHS} = \text{RHS}$ , $P(1)$ is true.

Step 2 (Assumption): Assume $P(k)$ is true for some integer $k \ge 1$ . i.e.,

$1 + 3 + \dots + (2k - 1) = k^2$

Step 3 (Inductive Step): Prove $P(k+1)$ is true. Required to prove:

$1 + 3 + \dots + (2k - 1) + (2(k+1) - 1) = (k+1)^2$

$\text{LHS} = [1 + 3 + \dots + (2k - 1)] + (2k + 1)$

Substitute the assumption from Step 2:

$\text{LHS} = k^2 + (2k + 1)$ $\text{LHS} = (k + 1)^2 = \text{RHS}$

Conclusion: Since $P(1)$ is true, and $P(k)$ implies $P(k+1)$ , by the principle of mathematical induction, $P(n)$ is true for all integers $n \ge 1$ .

6. Exam Strategy

Induction typically appears as a multi-part question in the Extension papers, usually positioned in the middle-to-late sections (Questions 12 to 14). Because the structure of the proof is rigidly defined, this is a prime area to demonstrate clear logical rigor. Even if the algebra in Step 3 is exceptionally difficult, you can almost always establish a solid foundation for your response just by correctly demonstrating the base case and explicitly stating the assumption.

7. Key Definitions / Glossary Summary

Base Case: The initial step proving the statement holds for the first value (usually $n=1$ ).
Inductive Hypothesis (Assumption): The assumption that the statement holds for an arbitrary integer $k$ .
Inductive Step: The algebraic proof showing that if it holds for $k$ , it must hold for $k+1$ .

8. Frequently Asked Questions (FAQ)

Q: Can I use abbreviations like "RTP" (Required to Prove) in the HSC? A: Yes, examiners accept standard abbreviations, but your final conclusion must be a complete, clearly communicated sentence.

Q: Are induction proofs for inequalities harder? A: Yes, inequalities often require transitive logic (showing $A > B$ and $B > C$ , therefore $A > C$ ) rather than simple, direct algebraic substitution.

9. Where to next?

Since induction relies heavily on recognizing patterns and constructing rigorous arguments, the natural next step is to tackle the HSC-Proofs booklet for Extension 2 students, or HSC-Sequences to apply these skills to arithmetic and geometric series.

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