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SingaporeFurther Maths

Mathematical Induction, Inequalities and Recurrences

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do we prove and solve inequalities rigorously, and which standard inequalities are worth knowing?

Prove and apply inequalities including the use of the discriminant, completing the square, and standard results such as the AM-GM inequality
A focused answer to the H2 Further Mathematics outcome on proving and applying inequalities. Algebraic manipulation, completing the square, the discriminant condition, the AM-GM inequality, and rigorous proof techniques.
10 min answer →

What are the standard methods of mathematical proof, and how do we structure a rigorous argument?

Construct rigorous mathematical arguments using direct proof, proof by contradiction, proof by contrapositive, and disproof by counterexample
A focused answer to the H2 Further Mathematics outcome on methods of proof. Direct proof, proof by contradiction, proof by contrapositive, disproof by counterexample, and the logical language of implication, converse and equivalence.
10 min answer →

How does proof by mathematical induction establish a statement for all positive integers, and how do we write a watertight induction argument?

Prove statements about sums, divisibility and inequalities for all positive integers using the principle of mathematical induction
A focused answer to the H2 Further Mathematics outcome on proof by mathematical induction. The principle, the base case and inductive step, and worked proofs for sums, divisibility and inequalities with the markers each step earns.
11 min answer →

How do we solve linear recurrence relations to find a closed form for the nth term?

Solve first- and second-order linear recurrence relations with constant coefficients and find closed-form expressions for the nth term
A focused answer to the H2 Further Mathematics outcome on linear recurrence relations. First-order and second-order constant-coefficient recurrences, the characteristic equation, repeated and complex roots, and particular solutions for non-homogeneous cases.
11 min answer →

How do we sum a finite series in closed form using the method of differences and the standard power sums?

Sum finite series using the method of differences, standard results for powers of integers, and partial fractions
A focused answer to the H2 Further Mathematics outcome on summing series. The method of differences (telescoping), the standard sums of powers of integers, partial fractions to create a telescoping form, and recovering sums to infinity.
10 min answer →