Game plan 2 · Unit 3

Induction and trigonometric proofs

One five-step structure takes you to four kinds of proof: summation, divisibility, De Moivre’s theorem and multi-angle identities.

Reach for it when

You see “prove by mathematical induction”, $\Sigma$ notation, “for all positive integers $n$”, “divisible by”, or “prove that $\cos3x=\dots$ using De Moivre’s theorem”.

Part 1

Read the notation first

A sequence is a list that follows a rule, and a series is that list added up. The QCAA gives the series in sigma notation and expects you to write it out before you prove anything.

$$\sum_{j=1}^{n}(2j-1)=n^2\quad\text{is}\quad1+3+5+7+\dots+(2n-1)=n^2$$

i, the lower limit i equals 1 is the first value, sigma means sum, and x sub i is the formula for the terms." style="display:block;width:100%;height:auto"> Σ n i = 1 xi Upper limitthe last value of i Sigmameans “add them up” Lower limitthe first value of i The termswhat is being added
Write the sum out before you touch the algebra. The upper limit is what changes from $k$ to $k+1$, and the term you split off is the key to the inductive step.

Watch for

Misreading the sigma is the most common way to lose a summation proof before it starts.

In the Topic Guides

Part 2

How the QCAA awards the six marks

This is drawn from the marking guides from 2020 to 2025. Most of the marks are for structure. If you know the five steps, you can earn them before the algebra starts.

The responseMark
Correctly proves the initial statement1
Correctly formulates an appropriate assumption1
Establishes the expression for the left-hand side of the inductive step1
Uses the assumption within the inductive step1
Completes the proof by reaching the right-hand side, or by factoring out the divisor1
Shows logical organisation, having attempted all steps, including a suitable conclusion1

Habit

Write the conclusion even if the working fell apart. It carries its own mark. In the assumption, always state the domain: “for $n=k$, where $k\ge1$”.

In the Topic Guides

Part 3

Summation proofs

Game plan 2.1

Proving a sum by induction

  1. 1Show it is true for $n=1$. Work out the left-hand side and the right-hand side separately, say that they are equal, and so it is true for $n=1$.
  2. 2Assume it is true for $n=k$, where $k\ge1$. Write out the whole assumed statement.
  3. 3Write the sum for $n=k+1$ using the assumption. Split it as $S_{k+1}=S_k+\text{the }(k+1)\text{th term}$, then replace $S_k$ with the assumption.
  4. 4Rearrange until it matches the right-hand side for $n=k+1$. Factorise early and use common denominators.
  5. 5Conclude by induction. The proposition is true for $n=k+1$ whenever it is true for $n=k$, and it is true for $n=1$. So by mathematical induction, it is true for all integers $n\ge1$.

In the Topic Guides

Part 4

Divisibility proofs

Game plan 2.2

Proving divisibility by induction

  1. 1Show it is true for $n=1$. Work out the expression and show it is a multiple of the divisor.
  2. 2Assume it is true for $n=k$: $f(k)=\text{divisor}\times Q$ for some integer $Q$.
  3. 3Write $f(k+1)$ and rewrite it so that $f(k)$ appears. Add and subtract the term you need, then replace $f(k)$ with $\text{divisor}\times Q$.
  4. 4Take the divisor out of everything that is left, so $f(k+1)=\text{divisor}\times(\text{an integer})$.
  5. 5Conclude by induction, with the same sentence as Game plan 2.1.

In the Topic Guides

Part 5

Proving De Moivre’s theorem

Game plan 2.3

Proving $(\text{cis}\,\theta)^n=\text{cis}(n\theta)$

  1. 1Show it is true for $n=1$. It is obviously true, but you still need to show it.
  2. 2Assume it is true for $n=k$: $(\text{cis}\,\theta)^k=\text{cis}(k\theta)$.
  3. 3Write the $n=k+1$ case using the assumption: $(\text{cis}\,\theta)^{k+1}=(\text{cis}\,\theta)^k\,\text{cis}\,\theta=\text{cis}(k\theta)\,\text{cis}\,\theta$.
  4. 4Expand into Cartesian form, group the real and imaginary parts, then use the compound angle identities to write it as $\text{cis}((k+1)\theta)$.
  5. 5Conclude by induction, with the same sentence as Game plan 2.1.

In the Topic Guides

Part 6

Multi-angle identities

These go up to $4x$. You don’t use induction here. You use De Moivre’s theorem and the binomial theorem together.

Game plan 2.4

Proving a multi-angle identity

  1. 1Write $(\cos x+i\sin x)^n$ two ways. De Moivre’s theorem gives $\cos nx+i\sin nx$, and the binomial theorem gives the expansion.
  2. 2Simplify the powers of $i$ in the expansion: $i^2=-1$, $i^3=-i$, $i^4=1$.
  3. 3Equate parts. Use the real parts for a $\cos nx$ identity and the imaginary parts for a $\sin nx$ identity.
  4. 4Convert any leftover mixed powers using $\sin^2x=1-\cos^2x$, or its partner, until only the function you need is left.

In the Topic Guides

Formulas

Formula panel

Each formula is labelled with how to treat it in the exam.

Compound angle identities for $\sin(A\pm B)$ and $\cos(A\pm B)$
On the sheet
The binomial theorem $(x+y)^n$, listed under Statistics
On the sheet
De Moivre’s theorem
On the sheet
$\sin^2A+\cos^2A=1$
On the sheet
The five steps and the conclusion sentence
Memorise
The cycle of powers of $i$
Memorise

QCAA language

What the question says, and what it means

Question typeWhat you’ll seeWhat it means
Summation$\Sigma$, “the sum of the first $n$ …”Game plan 2.1. Split off the $(k+1)$th term
Divisibility“divisible by 7 for all $n$”Game plan 2.2. Add and subtract
De Moivre power proof“prove $(\text{cis}\,\theta)^n=\text{cis}(n\theta)$”Game plan 2.3
Multi-angle“show $\cos3x=4\cos^3x-3\cos x$”Game plan 2.4. Equate the real parts

Back to the External Game plan or the Topic Guides.

Wording quoted from past QCAA Specialist Mathematics external assessments is © State of Queensland (QCAA), licensed under CC BY 4.0. Tangent Tuition is not affiliated with the QCAA.

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