Define P(n)
State exactly what is to be proved and its integer domain.
H3 MATHEMATICS · MATHEMATICAL INDUCTION
Anchor the first case.
Then build the logical bridge.
A proof-writing guide for Singapore A-Level H3 Mathematics syllabus 9820. State the proposition precisely, use the induction hypothesis visibly and close the argument for every required integer.
- Precise propositions
- Valid inductive steps
- Complete conclusions
Built around 9820 H3 MathematicsProof by mathematical induction and rigorous mathematical argument
A chain, not a patternThe inductive step must logically use P(k) to establish P(k+1)
THE LOGICAL ENGINE
Base case plus implication
Checking several cases can suggest a statement, but it cannot prove infinitely many cases. Induction combines one verified starting point with a universal implication.
The induction hypothesis is temporary and conditional: assume P(k) for an arbitrary valid k, then use it to prove P(k+1).
THE PROOF FRAME
Make every logical link visible
Verify the base
Substitute the first required value on both sides.
Assume P(k)
Take an arbitrary k in the domain and state the hypothesis.
Prove P(k+1)
Transform the next case, explicitly invoking the hypothesis.
WORKED EXAMPLE
Prove the sum of the first n odd integers.
Prove by induction that for every positive integer n.
EXAM HABIT
In the next case, append the new term; do not replace the whole sum without explanation.
- 1
Base case n=1
So P(1) is true.
- 2
Induction hypothesis
Assume for some positive integer k that
- 3
Build the next case
- 4
Conclude
Thus P(k) implies P(k+1). Since P(1) is true, the statement holds for all positive integers n by mathematical induction.
Proved1 + 3 + ⋯ + (2n − 1) = n² for all positive integers n.
COMMON MISTAKES
What to catch before the examiner does
Checking examples only
Several true cases suggest a result but do not prove all later cases.
Assuming P(k+1)
The induction hypothesis is P(k); P(k+1) is what must be established.
Not using the hypothesis
The inductive step must visibly depend on the assumed kth case.
Incomplete conclusion
Name the domain and invoke mathematical induction after proving both links.
PRACTISE WITH FEEDBACK
Have the logic—not only the algebra—checked.
Integrand can identify a missing base case, circular reasoning or an unused hypothesis.
Try Integrand- 1
State
Define P(n) and its domain.
- 2
Bridge
Use P(k) to prove P(k+1).
- 3
Conclude
Invoke induction precisely.
QUICK QUESTIONS
H3 mathematical induction FAQ
Why is the base case necessary?
The implication P(k)⇒P(k+1) can propagate truth only after at least one valid starting case has been established.
Can I assume the result for every k?
Assume it for an arbitrary k in the stated domain, only for the purpose of proving the next case.
Is this guide within the H3 syllabus?
Yes. Proof by mathematical induction is included in Singapore A-Level H3 Mathematics syllabus 9820.
READY TO PRACTISE?