Build an infinite chain of reasoning

Mathematical induction proves a starting case and an implication from an arbitrary case k to k + 1. Together they establish the claim for every integer from the starting point onward.

State exactly what is assumed at k; you cannot assume the result at k + 1. For a sum, add its next term to the induction hypothesis, then simplify to the target form. Complex expansions also prove trigonometric identities by comparing real or imaginary parts.

\displaystyle P(n_0),\quad P(k)\Rightarrow P(k+1)\quad\Longrightarrow\quad P(n)\text{ for all }n\ge n_0

WORKED EXAMPLE

Prove 1 + 3 + … + (2n − 1) = $n^{2}$ for positive integers.

At n = 1, both sides equal 1.
Assume the sum of the first k odd integers equals $k^{2}$ .
Adding 2k + 1 gives $k^{2}$ + 2k + 1 = (k + 1)².
The starting case and induction step establish the claim for all positive integers.

Assumed knowledge

Follow a direct proof, manipulate integer expressions and finite sums, and expand and factorise algebra. Recall complex multiplication when using real and imaginary parts to establish identities.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Build an infinite chain of reasoning

Prove 1 + 3 + … + (2n − 1) = n2n^{2}n2 for positive integers.

Put the idea to work.

Prove 1 + 3 + … + (2n − 1) = $n^{2}$ for positive integers.