Algebra, Number and Structure: Expressions, Equations and Sequences
Core concepts
This domain rewards precision with symbols. You must manipulate algebraic expressions confidently: expanding brackets, factorising common factors and quadratics, simplifying algebraic fractions, and applying index laws including negative and fractional indices. Know how to transpose formulas to make a nominated variable the subject, and how to solve linear equations, simultaneous linear equations (substitution or elimination), and quadratic equations by factorising, completing the square or the quadratic formula. Understand linear and non-linear recurrence relations, arithmetic and geometric sequences, and the difference between recursive and closed-form definitions. Ratio, proportion, percentage change, and financial mathematics such as simple and compound interest belong here too; know that compound growth uses repeated multiplication by (1 + r).
Traps to watch
- Sign errors when expanding negative brackets, e.g. −(2x − 5) = −2x + 5.
- Forgetting that x² = k has two solutions, x = ±√k, unless the context restricts the domain.
- Applying percentage increase and decrease in the wrong order; a 20% rise then a 20% fall does not return to the original value.
- Misreading recurrence notation such as tₙ₊₁ = 1.05tₙ as a simple-interest model.
- Cancelling terms rather than factors in algebraic fractions.
Worked example
Solve 2x² − 7x + 3 = 0. Factorising gives (2x − 1)(x − 3) = 0, so x = 0.5 or x = 3. Check by substitution: 2(0.25) − 3.5 + 3 = 0. For a recurrence t₁ = 500, tₙ₊₁ = 1.04tₙ, the closed form is tₙ = 500 × 1.04^(n−1), a geometric sequence with common ratio 1.04.
Rule of thumb
Always substitute your solution back into the original equation or relation; it catches most sign and arithmetic slips.