Mixed Proofs & Exam Technique
You know direct proof, induction, strong induction, and contradiction. Now the exam won't tell you which to use. This final lesson gives you a decision framework, two worked mixed proofs, and the exam habits that separate Band 6 answers from the rest. No new techniques — just sharper execution.
For each statement below, without looking anything up, write which proof technique you would use and briefly why.
(a) "Prove $n^3 - n$ is divisible by $6$ for all integers $n$."
(b) "Prove $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$ for all $n \geq 1$."
(c) "Prove $\sqrt{5}$ is irrational."
In the HSC, proof questions often do not name the technique. Read the statement structure first, then select.
Ask these questions in order:
1. Does the statement say "for all $n \geq 1$" (or a similar integer index)? → Induction
2. Does the inductive step require earlier cases (not just $k$)? → Strong induction
3. Is the claim a negative — irrational, impossible, infinite, unique? → Contradiction
4. Is it a direct algebraic or logical consequence? → Direct proof
Note: "for all $n$" does not always require induction — sometimes direct algebraic factoring is simpler. Choose the technique that keeps the proof shortest and clearest.
Key facts
- The four techniques: direct, induction, strong induction, contradiction
- When each technique is the most efficient choice
- The trigger phrases that signal each technique in an HSC question
Concepts
- Why "for all $n$" does not automatically mean induction
- The relationship between the statement structure and the proof method
- How examiners mark proof questions (structure, working, conclusion)
Skills
- Identify the correct technique for any Module 5 HSC question
- Carry out direct, induction, and contradiction proofs exam-ready
- Write proofs that earn full marks: assumptions stated, working shown, conclusion explicit
Train yourself to read these trigger phrases and automatically map them to a technique.
"Prove for all positive integers $n$…"
Trigger: integer index. Default: try direct first; if not obvious, use induction.
"Prove the following identity for all $n \geq 1$…"
Trigger: identity + integer index. Default: mathematical induction.
"Prove $f_n$ satisfies the recurrence… for all $n$"
Trigger: recurrence, depends on earlier cases. Default: strong induction.
"Prove that $\sqrt{m}$ is irrational"
Trigger: irrationality. Default: contradiction.
"Prove that there are infinitely many…" or "Prove there is no…"
Trigger: infinitude or impossibility. Default: contradiction.
Quick check: Which technique is most appropriate for: "Prove that $n^3 - n$ is divisible by $6$ for all integers $n$"?
Train yourself to read these trigger phrases and automatically map them to a technique.
Pause — copy the trigger-word-to-technique table (sums/products/divisibility → induction; irrational/no-solution → contradiction; "if P then Q" → direct or contrapositive) into your book.
Worked examples · 2 mixed proofs, technique chosen from scratch
Prove that $n^3 - n$ is divisible by $6$ for all integers $n$.
Prove by induction that $\dfrac{d^n}{dx^n}(x^{n-1}\ln x) = \dfrac{(n-1)!}{x}$ for all $n \geq 1$.
We just saw that trigger phrases like "prove for all $n$", "prove divisible by $d$", and "prove the $n$th derivative" map directly to induction, while "prove irrational" or "prove no real solution" call contradiction. That raises a question: beyond recognising the technique, what exam-room habits separate a B6 answer from a B4 answer? This card answers it → state proof structure explicitly, write the conclusion sentence, and use $k$ not $n$ in the hypothesis.
- State your assumptions. Write "Assume $P(k)$ is true" or "Suppose, for contradiction, that…" — these are often worth 1 mark each.
- Show all working. Examiners follow your logic step by step. A missing step cannot earn a mark.
- Conclude clearly. "Hence, by mathematical induction, $P(n)$ is true for all $n \geq 1$." or "This is a contradiction, therefore…". These closing sentences are frequently mark-awarded.
- Check base cases. An induction proof without a valid base case is not a proof. Do not skip it.
- Allocate time wisely. Proof questions carry 3–5 marks. Allow roughly 1 minute per mark. If you're in minute 6 of a 3-mark proof, reconsider the technique.
Did you get this? True or false: in an induction proof, omitting the base case means the argument is still valid as long as the inductive step is correct.
Did you get this? True or false: in an induction proof, omitting the base case means the argument is still valid as long as the inductive step is correct.
Pause — copy the three exam habits that protect marks on every proof question: (1) state proof structure, (2) write conclusion sentence, (3) use $k$ not $n$ in the hypothesis into your book.
Fill the gap: The correct closing sentence for an induction proof is: "Hence, by mathematical , $P(n)$ is true for all $n \geq 1$."
Misconceptions to fix · 4 common Module 5 exam errors
Did you get this? True or false: in the inductive step, you are allowed to assume both $P(k)$ and $P(k+1)$ to prove the result.
Activities · select the technique, then write the proof
State the best technique and carry out the proof: "Prove $n^3 - n$ is divisible by $6$ for all integers $n$." (Direct or induction?)
Use induction to prove that $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$ for all $n \geq 1$.
Prove by contradiction that $\sqrt{5}$ is irrational. Follow the four steps.
Identify the technique error: "We want to prove $P(n)$. Assume $P(k+1)$ holds. Then $P(k)$ must also hold because…"
Write the correct closing sentence for: (a) an induction proof, and (b) a contradiction proof.
Odd one out: Three of these statements are correctly matched to their best proof technique. Which pairing is WRONG?
At the start you chose techniques for three statements. For $n^3 - n \div 6$: the answer is direct proof (factor into three consecutive integers). For the sum of squares identity: induction. For $\sqrt{5}$ irrational: contradiction. Did your initial instincts align?
The biggest lesson from Module 5: the technique choice is part of the mathematical thinking, not just a formality. Selecting the right method and executing it cleanly is what earns Band 6 marks.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. Determine the most appropriate proof technique for "Prove $n^3 - n$ is divisible by $6$ for all $n$" and carry out the proof. (3 marks)
Q2. Prove by induction that $\dfrac{d^n}{dx^n}(x^{n-1}\ln x) = \dfrac{(n-1)!}{x}$ for all $n \geq 1$. (3 marks)
Q3. A student writes: "To prove $P(n)$ by induction, I assume $P(k+1)$ is true and then show $P(k)$ must hold." Identify the error and explain the correct approach. (2 marks)
Comprehensive answers (click to reveal)
Activity answers:
1. Direct proof: $n^3-n = (n-1)n(n+1)$. Consecutive integers $\Rightarrow$ one divisible by 2, one by 3, product by 6. $\square$
2. Base ($n=1$): $\sum_{k=1}^{1}k = 1 = \frac{1\cdot2}{2}$ ✓. Assume $\sum_{k=1}^{m}k = \frac{m(m+1)}{2}$. Then $\sum_{k=1}^{m+1}k = \frac{m(m+1)}{2} + (m+1) = \frac{(m+1)(m+2)}{2}$. By induction, true for all $n \geq 1$. $\square$
3. See worked example in the lesson.
4. Error: you cannot assume $P(k+1)$ in an induction proof. You assume $P(k)$ (the inductive hypothesis) and prove $P(k+1)$ follows.
5. (a) "Hence, by mathematical induction, $P(n)$ is true for all $n \geq 1$." (b) "This is a contradiction. Therefore [original statement] must be true."
Q1 (3 marks): [1] Identifies direct proof as most efficient. [1] Factors $n^3-n = (n-1)n(n+1)$. [1] Argues consecutive integers divisibility and concludes.
Q2 (3 marks): [1] Base case verified. [1] Inductive step correctly applies product rule and hypothesis. [1] Explicit conclusion naming mathematical induction.
Q3 (2 marks): [1] Error identified: assuming $P(k+1)$ is circular. [1] Correct approach: assume $P(k)$ is true, then derive that $P(k+1)$ must also be true.
Five timed questions spanning all Module 5 proof techniques. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). This is the module finale — give it your best.
⚔ Enter the arenaClimb platforms by answering mixed proof questions. Lighter alternative to the boss.
Mark lesson as complete
Tick when you've finished the practice and review. You've completed Module 5!