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 confidencethat 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!