HSC Induction: Study Guide & Outline

The introduction contextualizes the booklet, outlining the project overview, target audience (HSC Extension 1 and 2 students), and usage guide. A dedicated induction primer recapitulates the principle of mathematical induction: proving a statement for $n=1$, assuming it for $n=k$, and then proving it for $n=k+1$. It highlights common pitfalls like forgetting the base case or misapplying the inductive hypothesis, and sets the stage for the structured problem sets that follow.

This section presents fully worked solutions to a curated set of induction problems organised into Basic, Medium, and Advanced tiers. Basic problems ($2.1-2.6$) cover foundational skills like proving divisibility (e.g., $n^3+2n$ is divisible by $3$, $9 \mid 7^n+2^n$), summation formulas (e.g., sum of odd numbers, telescoping harmonic sum), and geometric patterns (triangular numbers, polygon interior angle sum). Medium problems ($2.7-2.18$) introduce inequalities (bounding Basel-type sums, weighted geometric sums, product of odd factorials vs powers of factorials, bounding sums of cubes), solving linear recurrences, calculus applications (derivative of $x^n$, reduction of tangent integral, logarithmic reduction to closed form), and more sophisticated algebraic manipulation. Advanced problems ($2.20-2.24$) tackle De Moivre's theorem, tiling a defective $2^n \times 2^n$ board, evaluating integrals from recurrences, integral coefficients in $\int_0^1 x^n e^x\,dx$, and deriving closed forms. Each solution meticulously demonstrates the inductive step, often linking induction to other mathematical ideas.

Offering a rich bank of practice problems, this section presents concise solutions and strategic hints. Basic problems ($3.1-3.8$) reinforce routine induction skills: sums of odd numbers, squares, geometric series, divisibility of $4^n-1$ by $3$, Bernoulli’s inequality $(1+x)^n \ge 1+nx$, De Moivre’s conjugate form, and logarithmic derivatives. Medium problems ($3.9-3.16$) extend to sum of cubes, tower inequality $n^{n+1} > (n+1)^n$, factorial comparisons, the postage stamp problem, and exponential versus polynomial inequalities. Advanced problems ($3.17-3.30$) delve into classic theorems: Binomial Theorem, Fermat’s Little Theorem, Vandermonde’s Identity, Wilson’s Theorem, and challenging sequences like nested radicals, recursive sequences with surds, trigonometric sums (sum of cosines, cosecant sum, arctangent sum), Fermat number products and series, and a Fibonacci telescoping series. The hints guide students through the crucial inductive leaps, fostering independent problem-solving.

The conclusion synthesizes the role of mathematical induction as a cornerstone of rigorous proof in HSC mathematics, encouraging students to reflect on the techniques mastered throughout the booklet. It emphasizes the importance of understanding the logical foundation behind induction and suggests further exploration of advanced problems to solidify competence. The final message underscores that proficiency in induction not only secures examination marks but also deepens appreciation for mathematical reasoning.